In large saucepan over medium-high heat, melt butter; add onion and carrot slices and cook and stir until just tender, about 3 to 5 minutes. Stir in flour and seasonings.
Add chicken broth and cook and stir as you bring mixture to a boil. Stir in halved Brussels sprouts; reduce heat. Cover and simmer until Brussels sprouts are just tender, about 7 minutes (may be shorter if using frozen thawed sprouts).
In a small bowl, beat together egg, milk and wine and stir into soup. Heat until soup returns to a boil.