Clean and wash the leek and cut into thin rings. Peel and quarter carrots and cut into small pieces.
Heat 1 tablespoon of oil in a large pot. Sauté the leek and carrots, up to 2 tbsp each, for 3-4 minutes, turning them over. Add wine, bring to the boil and simmer for 2-3 minutes. Add stock, bring to the boil again and simmer for about 15 minutes.
Grating cheddar cheese. Stir the cheddar and processed cheese into the soup halfway through the cooking time. Season soup with salt and pepper.
In the meantime heat 2 tablespoons of oil in a frying pan. Fry the minced meat and the rest of the vegetables for about 5 minutes while turning them. Season with salt and pepper. Wash chives, shake dry and cut into small rolls.
Add the minced meat mixture to the soup and sprinkle with chives.