Prepare homemade rough puff pastry dough through 2nd refrigeration. If using store-bought frozen puff pastry, make sure it’s thawed. Keep either dough in the refrigerator until step 4 below.

Over medium heat in a large skillet, cook butter and sliced shallots, stirring often, until shallots begin to soften, about 3 minutes. Add mushrooms, garlic, salt, pepper, thyme, and rosemary. Stir and cook until mushrooms are soft, about 5–6 minutes. Remove from heat and stir in balsamic vinegar. Set aside to slightly cool as you roll out the dough.

Preheat oven to 400°F (204°C). Line 2 large baking sheets with parchment paper or silicone baking mats. Set aside.

On a lightly floured work surface using a lightly floured rolling pin, roll pastry dough into a rectangle about 10×17 inches in size. The dough will be around 1/8-inch thick. (If using store-bought, place the edge of one sheet over the other and use a rolling pin to adhere them together. Roll the whole thing out into a 10×17-inch rectangle.) Using a pizza cutter or a sharp knife, cut down the middle so you have two 5×17-inch rectangles, then cut across so you have 10 rectangles, each about 3.5×5 inches in size.

Brush egg wash all over puff pastry rectangles. Crimp edges with a fork. Carefully transfer dough to lined baking sheets, 5 tarts per sheet. If dough loses shape, use your hands to reshape into rectangles. Sprinkle each with about 1.5 Tablespoons cheese. Spoon the mushroom topping, about a scant 1/4 cup for each tart, over the cheese. Lightly press down on mushroom topping with the back of a spoon so it sticks to the cheese.

Bake tarts for 20–22 minutes, or until the pastry is golden brown all over.

Remove from the oven and, if desired, garnish with flaky sea salt, more fresh herbs, a little extra cheese, and/or a drizzle of balsamic glaze. Serve warm or at room temperature.

Cover and store leftovers in the refrigerator for up to 5 days. Reheat in the microwave, or cover with foil and bake in a 300°F (149°C) oven for 8–10 minutes.