In a medium skillet, melt 1 tbsp butter. Add chicken and cook until browned and cooked through.
Add onion, celery, carrot, and broccoli and cook until the vegetables are tender.
Remove chicken and vegetables to a plate and keep warm.
Heat 3 tbsp. butter in the same skillet. over medium heat. Add flour, thyme, parsley, bay leaf, salt, and pepper. Cook and stir for 1-2 minutes.
Add chicken broth, whisking constantly. Cook and stir over medium-high heat until boiling. Cook and stir for one minute more. Add cream.
Return chicken and vegetables to the skillet with the sauce.