Cheesy Mushroom Tart with Eggs

Ingredients

• 1 sheet puff pastry
• 2 eggs
• 9 ounces mushrooms (cut into large pieces)
• ½ cup grated Pecorino (Parmesan or mild Cheddar)
• ½ cup grated mozzarella
• ¾ cup half-and-half
• 1 shallot (finely chopped)
• 4 garlic cloves (minced)
• 3 tbsp olive oil
• 1 tbsp flour
• 2 tbsp balsamic vinegar
• ¼ tsp nutmeg
• 1 tbsp chopped fresh thyme
• salt (to taste)
• black pepper (to taste)
• 1 egg beaten with 1 tsp milk (for brushing)