How to Make It
Preheat oven to 375°F. Place cheese in freezer until ready to use (this will make it easier to shred). Toss together parsley, shallot, and lemon zest in a small bowl. Stir in 1/4 cup of the oil and 1/4 teaspoon of the salt. Set gremolata aside.
Heat butter and remaining 1 tablespoon oil in a large skillet over medium-high until butter is melted and mixture is sizzling, about 1 minute. Add mushrooms and spread in an even layer. Cook, undisturbed, until mushroom bottoms are browned, about 5 minutes. Stir mushrooms. Continue cooking, stirring occasionally, until browned all over, about 5 minutes. Stir in 1/2 cup gremolata and remaining 1/4 teaspoon salt. Cook, stirring constantly, until fragrant, about 10 seconds. Remove from heat.
Lightly grease a 12-cup muffin pan with oil. Drape 1 prosciutto slice inside each muffin cup, folding and overlapping each slice as needed to cover bottom and sides of cup, and letting edges of prosciutto hang over cup edges.
Remove cheese from freezer, and shred using large holes on a box grater. (You’ll have about 1 1/4 cups shredded cheese.) Sprinkle half of the cheese (about 2/3 cup) evenly into prepared muffin cups (about scant 1 tablespoon per cup). Top cups evenly with mushroom mixture (about 1 scant tablespoon per cup), pressing into bottoms of cups. Crack 1 egg into each cup. Spoon 1 teaspoon cream over each egg. Sprinkle evenly with remaining cheese. Sprinkle with desired amount of salt and pepper.
Bake until egg whites are just set, 12 to 15 minutes. Remove from oven; sprinkle eggs evenly with remaining gremolata (about 1 teaspoon each). Let cool 5 minutes.
Run an offset spatula or a butter knife around egg cup edges to loosen; remove from pan. Serve immediately.