Caramelized Onion and Mushroom Tart

white mushrooms, washed, dried, and sliced

unsalted butter, divided

large onions, peeled, halved, and thinly sliced

sprigs fresh thyme, leaves only

granulated sugar

garlic, finely minced

salt, plus more to taste

few grinds fresh black pepper

grated cheese (I used Comté, but really anything mild would work here, even cheddar or mozzarella - just don't used Parmesan as it will be too salty)

package frozen puff pastry, thawed

egg, beaten