Preheat oven to 350 degrees with rack in center position.
Drizzle 1 tablespoon oil along bottom of an 8-by-8-inch or similar baking dish. Place mushroom caps upside down in baking dish in a single layer.
In a large saute pan heat 1 tablespoon oil and butter over medium-high heat. When oil shimmers, add mushroom stems and salt and saute until mushrooms release moisture and become golden brown, 5 minutes.
Add garlic, red-pepper flakes, and parsley to pan and continue cooking, 30 seconds. Add lemon zest and breadcrumbs to pan and cook, stirring frequently, until breadcrumbs become toasted and golden, about 2 minutes.
Transfer mixture to a heatproof medium bowl and stir in lemon juice, Parmesan, and egg whites. Divide mixture between mushroom caps, transfer to oven, and bake until mushrooms darken and shrink, 25 to 30 minutes.
Carefully transfer mushrooms to a serving platter, garnish with grated Parmesan, and serve warm.