Cook vegetables in butter in a 5- to 6-quart heavy pot, covered, over medium-high heat, stirring occasionally, until beginning to brown and stick to the bottom of the pot, about 15 minutes.
Add chicken stock, 1 ½ teaspoons salt, and ¼ teaspoon pepper and simmer, covered, until vegetables are tender, about 10 minutes.
Transfer 3 cups of the soup to a blender or (food processor) with milk and blend until smooth. (Be sure to use caution when blending hot liquids).
Return to pot, then stir in dill and salt and pepper to taste.