In a medium mixing bowl, crack in the eggs. Add the heavy cream, shredded cheddar cheese, black pepper, and salt if using.
Whisk this 1 mixture together thoroughly until the eggs, heavy cream, cheddar, and black pepper are well combined and the cheese is evenly distributed. The mixture should look smooth with little shreds of cheese throughout.
Place the prepared muffin tray on a baking sheet (this makes it easier to move in and out of the oven). Carefully pour the whisked egg, cream, cheddar, and pepper mixture into each ham-lined muffin cup, filling each about 3/4 full. If any cheese clumps, gently nudge it so each cup gets some.
Bake in the preheated oven for 15–18 minutes, or until the egg mixture is set in the center and the tops look slightly puffed and lightly golden. The ham edges may get a little crisp, which adds nice texture.
Remove the muffin tray from the oven and let the breakfast cups cool in the pan for about 5 minutes. This helps them firm up and makes them easier to lift out.
Run a small knife or spatula around the edges if needed, then gently lift each ham and egg breakfast cup out of the pan. Serve warm, or let them cool completely before storing in the refrigerator for quick breakfasts later in the week.
