"NOWHERE ELSE IN the world does business like us," says David Greco, leaning over the high counter of his deli and handing me a fresh ball of exemplary mozzarella. "You can keep your fancy looks in Manhattan, we are all about quality."

On the strength of the cheese that I'm currently scoffing, he's not wrong. I could be scarfing it down in Campania, Italy, when in fact I'm in the Arthur Avenue Retail Market in the Bronx, New York City's most overlooked borough.

Built on the orders of NYC's most famous mayor, Fiorello La Guardia, the market has stood in its present building since 1940, catering to the Italian community which moved here to build the nearby Bronx Zoo. Today, it's the cornerstone of the area's Little Italy. While Manhattan's Italian streets are all but gone, Arthur Avenue remains a bastion of the old country. Its shops are run by third- and fourth- generation immigrants sticking hard and fast to traditional methods brought across the Atlantic by their forefathers.

David's spot, named Mike's Deli after his father who set up the business, is just one example. As I munch on an aubergine parmigiana, my guide, Susan Birnbaum, points out the market's other key joints. There are fresh produce stalls, alongside stands selling live seafood and a beer hall selling local Bronx beers set up by David. Anything Brooklyn can do, it seems, the Bronx can do better – and it doesn't stop there.

Susan is a Bronxite born and bred and still lives in the south of the borough, so she's the perfect person to take me around a part of New York that's easily left off of many tourists' itineraries.

Based on the strength of the cheese, I could be in Italy, but in fact, I'm in the Bronx

We make our way out of the retail market and walk a couple of blocks to East 187th Street, dropping into Borgatti's, where Chris, the owner, stands back from his pasta slicing machine to shout hello. Chris is third-generation Italian and his kids are already being trained up in the family business. Elizabeth, his daughter, is on social media duties, spreading the word about their first-rate ravioli, which a local Mama is raving about in Italian as we wait in line. Huge tresses of fresh pasta sit drying on racks out the back and I grab a bag of dried linguine before we head for the door.

As we continue our tour, I find the sheer breadth of bakeries, delis and restaurants in this part of the Bronx astonishing. You could easily spend an entire week eating here. With that in mind, and with my appetite showing no signs of abating, Susan leads me into Terranova, where they bake their bread in a vast, coal-fired oven. She snaffles us a loaf of lard bread, which is every bit as fatty and delicious as it sounds. The crusty exterior gives way to a gooey middle, laden with molten prosciutto. We each tear off a couple of pieces before Susan stuffs it back in her bag.

"I don't want other bakeries round here to see I've been buying up Terranova's treats," she laughs. That's not all I'm worried about. All that bread is sitting heavy and there's more eating to be done.

Our next stop is the wondrous Calabria Pork Store. As Susan pulls open the door of this unassuming joint, the smell smacks me in the face. I look up and see hundreds of cured sausages swinging from the ceiling. I take a small slice from the counter and lose myself in an epicurean reverie as it melts in my mouth.

There's no time to stick around though. After grabbing a slice of burek, a spanakopita-like treat, from Tony and Tina's, an Albanian-run joint at the top of Arthur Avenue, we head to Roberto's for an authentic Italian meal. Roberto himself is at the bar. Grey hair slicked back, decked out in clogs and chef's clothing, he greets us enthusiastically, buying us coffee and raving about the local ingredients. He takes us through his spectacular menu, from cavatelli with broccoli and Italian sausage to chicken with hot cherry peppers and white wine sauce.

I find the sheer breadth of bakeries, delis and restaurants astonishing

Feeling stuffed after our whistle-stop tour, I ask Susan where her favourite place to eat around here is. "Food is so personal," she says. "I can't pick favourites. You just can't get bad food here."

Having tasted some of the Bronx's Italian treats, I'm keen to see what Queens has to offer. One of the most ethnically diverse places on earth, the 'borough of nations' is not short of good food, but its Asian cuisine is renowned. I head to New World Mall in Flushing. A shopping centre food court might not strike you as prime foodie territory but the huge Asian community, which has boomed in Flushing since the 1970s, has made it the area's go-to hotspot.

I take a seat and marvel at the finest Asian food I've ever eaten outside the continent

After a scout around, I opt for pork belly and pickled green buns from Xiao Yuan Huang; takoyaki with wasabi mayo from the sole Japanese food stand; and pork, shrimp and sea cucumber dumplings from a Chinese Korean spot with little English signage. Everything is made to order. I take a seat and marvel at the finest Asian food I've ever eaten outside the continent. The whole meal comes to just $15.

I'm full to bursting, but need to make room for more. There's Greek food to eat in nearby Astoria, after all. Astoria's Greek population is writ large across the bars, clubs and restaurants which line the route beneath the elevated subway tracks in this west Queens neighbourhood. My first stop is BZ Grill for a chicken gyro, like a fancy kebab. The flatbread that's baked in the window is tempting me in, and it's a million miles from the British post–pub favourite. Succulent chicken; fresh salad; and a secret sauce our waiter refuses to divulge the ingredients of.

Almost defeated, I head north for some sweet treats at Artopolis Bakery. Its reputation precedes it, with numerous Greek Americans raving about the baklava and galatoboureko, a custard tart. They're not wrong. But after that gyro, I've no room left, so I get my order boxed up and head to the subway. I think of Susan's wise words about food and she's right, I can't pick a favourite either. But next time I'm in New York, I'll be leaving Manhattan behind when it gets to lunchtime.

Susan Birnbaum runs regular food tours around the Bronx, Queens and Lower Manhattan. Tours start from $49 for a three-hour session. Visit susansez.com for more details or to book onto a tour