1. In a skillet over high heat, melt the butter and gently stir-fry the livers 5 minutes; remove with slotted spoon and keep warm.
2. Add flour to the skillet, stir into butter, and cook, stirring, 1 minute.
3. Slowly add the broth, whisking constantly, until you have a thick sauce.
4. Lower the heat, add the next 5 ingredients (onion through pepper) and simmer 3-4 minutes, stirring constantly.
5. Add the Madeira, stir into the mixture and simmer 5 more minutes.
6. Stir in chicken livers and heat through.