It's Monday lunchtime and Steff's Place is packed. A simple chrome kiosk with a brightly coloured menu opposite the luggage carousels in terminal three of Copenhagen's Kastrup airport, it's a local institution. The reason? As the Danish diaspora returns home, even before they pick up their suitcases their thoughts turn to one thing: hot dogs.

The Danes are the world's most dedicated devourers of hot dogs with more than 50 different kinds of sausages available, from the humble røde pølser to the exotic-sounding kaempe knaek. Around 50 million sausages are produced for the fast-food market each year by leading Danish manufacturer Tulip, and Copenhagen is home to hundreds of pølsevognen (sausage wagons), many manned round the clock for the quintessential Danish experience of a hot dog on the way home after a night on the Carlsberg.

I'd come to the Danish capital to explore the national obsession with hot dogs and my first mouthful at Steff's Place piqued my appetite to know more. The simple hot dog of sausage, ketchup or mustard and fried onions, washed down with a cold local lager, tasted divine. Despite all the sleek eateries and stylish cafes that typify the burgeoning Danish gourmet scene, nothing could beat a dog at Steff's Place.

That was, however, until May this year when the Løgismose Grill bar opened and had the cognoscenti positively drooling at the prospect of a place where man and sausage could live in perfect harmony. Løgismose is part of Nimb, a new boutique hotel and restaurant in the city's historic Tivoli Gardens - the first accommodation actually inside Tivoli since the gardens opened in 1843. With its 13 rooms, a fine-dining restaurant, in-house deli and the cellar-bar Vinotek for tapas and wine, it oozes effortless Danish style.

So do the hot dogs. 'I got the idea from the pølsevognen across the road outside the central train station. I wanted to make hot dogs that were pure and high quality,' says head chef, Thomas Herman, giving me a crash course in the art of preparing the perfect Danish hot dog.

He insists the secret is to cook every-thing fresh on the grill. He has the smoked sausages specially prepared by a butcher in southern Jutland and the soft-dough brioche rolls prepared by a Japanese bakery. In a nod to the gastro-science of Heston Blumenthal, the onions are fried in duck fat and rosemary to add depth to the flavour, while the remoulade, the tartar-like sauce accompaniment, is prepared from a marinade of mushrooms and pickled vegetables. Hot dogs are traditionally washed down with chocolate milk, flavoured with vanilla, and Herman serves them with milk produced by the hotel's own organic dairy.

I'm about to tuck in when Thomas, who plans to expand the brand into a chain of gourmet hot-dog grills around the country, calls for a moment's contemplation of our creation.

'The perfect hot dog has three elements: the smoked flavour of the sausage, the sweetness of the sauces and the cinnamon-flavoured bread, and the sharpness of the onions,' he says.

'We grow up with hot dogs in Denmark. I remember going ice-skating with my father and we would always grab a hot dog on the way home. Hot dogs taste of nostalgia.'

After my class, I take a brisk pre-dinner constitutional around Tivoli with its mix of family-friendly attractions, gardens and 40-odd places to eat, ranging from organic frankfurters at Hot Dog Corner to The Paul, a Michelin-starred eatery run by British-born chef, Paul Cunningham. A ride on the Flying Carpet, named after the story by the city's most famous adopted son, Hans Christian Andersen, is the essential way to work up an appetite.

Dinner that night is simple, home-cooked Danish fare - a piece of succulent beef, preceded by six oysters, at the hotel's Restaurant Nimb. Seated on a high chair, I have a view of the chefs at work to my left and a vista across the gardens, glazed by hazy, summer sunshine, to my right.

But, before retiring to my room, I can't resist one last taste of hot-dog perfection. After a drink in the candlelit bar with its huge murals by the artist Cathrine Raben Davidsen, I sneak next door to the Løgismose Grill bar for one final bite. The bread is melt-in-the-mouth soft, the remoulade tantalisingly tangy and the sausage smoked to perfection. Steff's Place was good, but I have now joined the rarefied ranks of the hot-dog connoisseur - and found a little piece of hot-dog heaven.

• SAS Scandinavian Airlines (flysas.com) flies to Copenhagen from Heathrow, London City, Birmingham, Manchester and Aberdeen from £118 return. Nimb (00 45 88 700000; nimb.dk) has rooms from £425, including breakfast. Tivoli (00 45 3315 1001; tivoli.dk) has a daily programme of summer events until 21 September, Hallowe'en events from mid-October and a Christmas market from 14 November. More information from Wonderful Copenhagen (visitcopenhagen.com) and Visit Denmark (visitdenmark.com).