Preheat oven to 425 degrees. Grease 24 regular size muffin cups.
Mix hash brown potatoes, salt, black pepper, melted butter, and 2/3 cup shredded Cheddar cheese in a bowl. Divide mixture equally between prepared muffin cups, using your fingers or the back of a spoon to shape them into thin, hollowed-out nests. Tops edges can be uneven. Bake for 15-20 minutes or until the nests are brown around the tips and golden on the inside. Remove from oven.
Whisk eggs in a bowl and season with salt and pepper. Place 3-4 strips of ham into each nest, followed by the egg mixture. Sprinkle with red pepper and Cheddar cheese, and top with chives.
Bake in the oven until eggs are set, 10 to 12 minutes. Remove from oven and let cool for about 5 minutes. Remove by gently sliding a knife between the crust and the muffin cup. Serve while warm.