Preheat oven to 350 degrees F (175 degrees C). Grease a large 6-cup muffin tin with cooking spray.
Mix potatoes, melted butter, salt, and pepper together in a large bowl.
Divide potato mixture among muffin cups. Press to create a thin crust over the bottom and up the sides of each cup.
Bake in the preheated oven until edges are a light golden brown and potatoes are tender, about 30 minutes. Crack an egg gently into each cup. Continue cooking until egg whites are set and yolk is runny, about 6 minutes. Remove from the oven; cover eggs with provolone cheese.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Toss prosciutto with maple syrup and pepper in a small bowl. Spread out on a small baking sheet lined with aluminum foil.
Place muffin tin and baking sheet under the preheated broiler. Broil until cheese is melted and prosciutto is crisp, about 1 minute.
Transfer egg baskets to a serving plate. Chop prosciutto and sprinkle on top.
Substitute 3 cups shredded hash browns for the potatoes if desired. If using frozen, make sure they are thawed. Use a piece of cheesecloth or a clean kitchen towel to remove and squeeze out the moisture.
Substitute the breakfast meat of your choice for the prosciutto. Substitute Cheddar cheese for the provolone cheese if desired.
Per Serving: 399 calories;21.8 g fat;
34.5 g carbohydrates;
17.1 g protein;
230 mg cholesterol;
625 mg sodium.