In a large saucepan, melt the butter over medium-high heat. Add the chopped leeks, onion, carrot and garlic. Saute the vegetables; stirring occasionally, until they are glossy and have started to soften - about 5 minutes.
Stirring continuously, sprinkle in the flour. Still stirring, gradually pour in the chicken broth. Add the potatoes. Bring the liquid to a boil. Reduce heat to maintain a simmer and cook, partially covered, until the potatoes are tender enough to be pierced easily with a table fork, about 15 minutes.
Place the shredded cheese in a large heat proof mixing bowl. Ladle in about a quarter of the hot liquid from the soup and stir with a sturdy wooden spoon until the cheese has completely melted and is fully blended inches Carefully pour the cheese mixture back into the saucepan and stir until fully incorporated.
Stirring continuously, gradually pour in the heavy cream and add the salt and pepper. Continue to heat the soup until it almost reaches the boiling point. Turn off the heat, taste the broth, and adjust the seasonings if necessary with a little more salt and pepper. Ladle the soup into heated bowls. Garnish each bowl with the chopped corned beef and the chopped chives. Serve immediately.