Melt 1 T butter in a skillet over medium heat. Sautee onion and garlic about 5 minutes. Set aside.
Whisk 1/4 c melted butter and flour together in a large saucepan ofer medium heat, cooking 3-4 minutes. Add 1-2 T of milk to keep flour from burning.
Pour milk into flour mixture, gradually while whisking. Add chicken broth. Bring to a simmer and cook until it becomes a thick gravy texture, about 20 minutes. Add onion mixture, vegetables and seasonings. Simmer until vegetables are tender, about 20 minutes.
Stir cheddar cheese into soup until the cheese melts. Add salt & pepper to taste.