1. Place butter in heavy skillet. Add ham and onion. SautÚ over low heat until tender (approximately 2 minutes). Set aside.
2. Slice each bagel into 4 thin slices. Arrange 6 bagel slices at bottom of lightly greased baking dish. Top bagels with ham and onion mixture. Then top with cheddar cheese. Finally top with remaining bagel slices.
3. In a separate bowl whisk together eggs, milk, parsley, and salt and pepper to taste. Pour the egg mixture over the bagel layers.
4. Bake at 375F for approximately 30 minutes or until eggs are firm and cooked through.