Melt butter in a large saucepan or Dutch oven over medium-high heat. Cook and stir broccoli, mushrooms, carrot, and onion in hot butter until vegetables are tender, 6 to 8 minutes.

Stir in flour, basil, and pepper until combined. Pour in milk and half-and-half all at once; add Worcestershire sauce and bouillon granules. Cook and stir until thickened and bubbly. Add chicken; continue cooking until heated through.