Thaw puff pastry according to package. Preheat oven to 375 degrees F. In a medium bowl, whisk together eggs with milk.
In a large skillet or cast iron pan, add a drizzle of oil over medium heat. Once hot, add mushrooms and cook until they have lost their moisture and shrunk down. Then add garlic, thyme, red pepper flakes, and spinach. Cook for a minute or two until spinach is just wilted. SEason with salt and pepper and set aside.
Line a 13x9 sheet pan with parchment paper. Gently roll out puff pastry sheets on a lightly floured surface. Transfer sheets to sheet pan and crimp together in the middle to form a uniform piece of crust. Press edges of the crust up the sheet pan to form a thin crust for the fillings.
Spread out half of the grated cheese on the puff pastry, then sprinkle the mushroom/spinach mixture over the puff pastry. Top with the rest of the cheese.
Pour the egg mixture into the crust and use a spatula to distribute eggs evenly.
Bake the frittata until the eggs are just set in the center and the edges are golden browned and puffed, about 20 minutes.
Let frittata cool briefly and then slice into big pieces. Serve garnished with fresh thyme and hot sauce.