In a 5- to 6-quart Dutch oven heat butter over medium heat until melted. Add carrots, celery, and onion; cook for 6 to 8 minutes or until onion is tender and beginning to brown, stirring occasionally. Stir in bay leaves and, if desired, caraway seed; cook and stir about 1 minute or until toasted. Add broth. Bring just to boiling.
Meanwhile, for spaetzle, in a blender combine flour, the water, egg, milk, and salt. Cover and blend until smooth.
Fill a heavy resealable freezer bag with spaetzle batter; snip a small hole in one corner of the bag. Drop batter in about 1/2-teaspoon amounts onto hot bubbling soup. Simmer, uncovered, for 5 minutes. (Spaetzle will sink to the bottom and then rise to the top as they cook.)
Stir in kale and chicken. Simmer, uncovered, for 5 minutes more. Remove and discard bay leaves. Stir in lemon juice to taste.