Breakfast Egg Cups with Parsley Gremolata and Mushrooms

Name: Breakfast Egg Cups with Parsley Gremolata and Mushrooms
Ingredients: 5 ounces Taleggio or Brie cheese, rind removed; 1 cup loosely packed fresh flat-leaf parsley leaves, finely chopped; 3 tablespoons finely chopped shallot (from 1 shallot); 2 teaspoons grated lemon zest (from 1 lemon); 1/4 cup plus 1 tablespoon extra-virgin olive oil, divided, plus more for greasing; 1/2 teaspoon kosher salt, divided, plus more to taste; 1 tablespoon unsalted butter; 1 (8-ounce) package cremini or white button mushrooms, very thinly sliced (3 cups); 12 prosciutto slices (about 8 ounces); 12 large eggs; 1/4 cup heavy whipping cream; Black pepper, to taste