In a large pot, melt the butter over medium heat. Add the diced onion and cook until softened, about 5 minutes. Add the minced garlic and cook for another minute until fragrant.
STEP 2
Add the cooked chicken, diced ham, and chicken broth to the pot. Stir well, and bring to a gentle simmer.
STEP 3
Stir in the milk, heavy cream, thyme, salt, and pepper. Let the soup simmer for about 10-15 minutes to allow the flavors to meld.
STEP 4
Gradually add the shredded Swiss and mozzarella cheese, stirring until melted and smooth.
STEP 5
For a thicker soup, mix the cornstarch with 2 tablespoons of cold water until smooth. Add to the pot, stirring well, and cook for an additional 5 minutes until the soup thickens.
STEP 6
Ladle the soup into bowls, and garnish with extra cheese, a sprinkle of fresh herbs, or croutons if desired! Enjoy warm and creamy flavors with every spoonful.