Cheesy Eggplant Casserole

1 medium eggplant
3/4 cup soft breadcrumbs
2 tablespoons butter
2 tablespoons flour
1 cup milk (at room temperature)
2 teaspoons grated onion
1 tablespoon catsup
1 teaspoon salt
1/2 teaspoon pepper
Dash of cayenne
1/2 teaspoon Worcestershire sauce
6 ounces American or Cheddar cheese,
    cut into small pieces
2 eggs, slightly beaten (at room temperature )

Preheat oven to 350 degrees F. Butter a 1 1/2-quart casserole dish.

Peel and dice eggplant. Steam or saut