Preheat the oven to 400 degrees F (200 degrees C). Grease 2 mini muffin tins with melted butter; set aside.

Trim tough ends from fresh asparagus spears. Bring a large pot of salted water to a boil; cook asparagus in boiling water just until bright green, 1 to 2 minutes, then transfer into a bowl of cold water to stop the cooking process. Drain very well.

Cut off the asparagus tips and set aside. Slice enough of the remaining spears into 1/4-inch pieces to measure about 3/4 cup pieces. Reserve any leftover asparagus for another use.

Place 4 or 5 pieces of cut ham into the bottom of each muffin cup. Top ham with about 1 rounded teaspoon of asparagus pieces. Top asparagus with 1 teaspoon of grated cheese.

Whisk eggs, cream, salt, black pepper, and cayenne together very thoroughly, being sure that all the egg whites have been completely mixed in. Transfer into a pourable cup or container.

Fill each mini muffin cup to the top with egg mixture. There may be some egg mixture leftover depending on how much ham, asparagus, and cheese were added to each cup. Top each cup with 1 or 2 reserved asparagus tips; add another teaspoon cheese on top. Dust tops with cayenne if desired.

Bake in the preheated oven until egg bites are lightly golden brown on top, and egg is just set, about 18 minutes.