Preheat the oven to 400°F

Cut the button mushrooms into smaller pieces or medium slices. Don't cut the mushroom too thinly. Mince garlic.

Heat butter in a large frying pan over medium-low heat. When hot add the mushrooms. Cover and cook without stirring for 1-2 minutes, allow to brown. Throw in minced garlic, stir to cook. Then add the vinegar and cook. Mushrooms should have reduced their volume in half and all the liquid is evaporated. Don't stir the mushrooms for about a minute when almost all liquid is evaporated, letting the mushrooms brown a little.

Take the puff pastry out of the fridge. (I plop bot hpeices next to each other on a pan and smoosh into one rectangle and top) If your puff pastry is square when you buy it, roll it out slightly into a rectangular shape. With the tip of a sharp knife, score through the top layers of the puff pastry by running the knife along all four sides of the dough (leaving about 1-inch / 2.5 cm border). Do not cut all the way through the dough. Transfer the pastry, along with the parchment paper that comes with it to a baking sheet. Prick the dough (but not the borders) with a fork.

Sprinkle the cheese in an even layer over the dough.

Top with cooled mushrooms.

Brush the edges of the tart with beaten egg. Prick the dough with a fork a couple times more (or it will puff up in the oven), not the edges, just the center of the tart.

Bake the tart for about 20-25 minutes or until golden.