Cheesy Mushroom Tart with Eggs

Name: Cheesy Mushroom Tart with Eggs
Ingredients: 1 sheet puff pastry; 2 eggs; 250 grams (9 oz) mushrooms, cut into large pieces; 60 grams (1/2 cup) grated Pecorino; 60 grams (1/2 cup) grated mozzarella; 160 milliliters (3/4 cup) half-and-half; 1 shallot, finely chopped; 4 garlic cloves, minced; 3 tablespoons olive oil; 1 tablespoon flour; 2 tablespoons balsamic vinegar; 1/4 teaspoon nutmeg; 1 tablespoon chopped fresh thyme; salt + black pepper; egg wash