Preheat oven to 350 degrees F.
Boil chicken until no longer pink inside. Cut into 1 inch cubes.
Combine chicken, corn, and peas in a 13x9x2-inch greased baking dish.
In a small saucepan, melt butter over medium heat. Add onion and cook until soft, about 3 minutes. Blend in flour, salt, and pepper. Cook 1 to 2 minutes without browning. Gradually whisk in chicken broth and milk. Bring to a boil and cook, stirring constantly, until thick and smooth.
Remove from heat and stir in eggs. Pour sauce over chicken mixture.
Combine bread crumbs and cheese in a small bowl and sprinkle on top of casserole. Cover with foil. Bake for 20 minutes. Uncover and bake 10 minutes, until lightly browned on top.