1Pound cutlets with the site of a heavy knife or cleaver, between sheets of waxed paper, until very thin.
2In a large skillet over medium-high heat, melt 4 tablespoons butter and cook cutlets for about 1 minute per side or until nicely browned. Remove to a warm serving dish and keep warm.
3For the sauce, in the same pan, add beef broth and cook over high heat until reduced to 1/2 cup. Mix butter and flour into a paste and add to broth and stir until smooth. Keep hot.
4For the eggs, combine eggs, chives and salt and pepper. In another skillet over medium heat, add 4 tablespoons butter and saute mushrooms for 3 minutes, stirring constantly.
Add eggs and stir with a wooden spoon until eggs are barely set.
5Spoon eggs over cutlets. Streak each serving with the sauce and serve.