Preheat oven to 425°. In a large bowl, toss hash browns with oil, flour, 1 tsp. salt, and 1/2 tsp. pepper. Butter and flour two 6-cup jumbo muffin pans and divide hash browns evenly among cups, pressing into the bottoms and up sides of cups.
Bake until hash browns are beginning to brown on edges, 30 to 35 minutes. Reduce oven temperature to 325°. Sprinkle about half of prosciutto and gruyère evenly among nests. Crack an egg into each nest, then sprinkle remaining prosciutto and gruyère over the tops of eggs. Sprinkle with parsley and salt and pepper to taste. Bake until eggs are just set, 15 to 18 minutes. To serve, gently slide each nest out of the pan with a knife. Set on plates and serve with hot sauce if you like.