Combine chicken and half-and- half in a Dutch oven; cook over low heat 10 minutes, and set aside
Melt butter in a heavy saucepan over low heat; add flour, and stir until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until thickened and bubbly.
Stir 1/4 cup of the hot white sauce into egg yolks; add to remaining hot sauce mixture, stirring constantly. Stir in salt and pepper.
Add half of sauce to chicken mixture in Dutch oven; stir well. Pour chicken mixture into a well-greased 2-quart casserole. Sprinkle mushrooms evenly over chicken mixture. Add onion and 1/4 cup cheese to the remaining sauce; stir well, and pour over mushrooms. Sprinkle with remaining cheese. Broil 6 inches from heat for 7 minutes or until top is lightly browned.