Prepare homemade rough puff pastry dough through 2nd refrigeration or ensure store-bought dough is thawed, keeping it refrigerated until step 4.

In a large skillet over medium heat, 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.

Preheat oven to 400°F (204°C). Line 2 large baking sheets with parchment paper.

On a lightly floured work surface, roll pastry dough into a rectangle about 10x17 inches. Cut the dough into two 5x17-inch rectangles, then cut across to make 10 rectangles, each about 3.5x5 inches.

Brush egg wash all over puff pastry rectangles. Crimp edges with a fork. Transfer dough to lined baking sheets, 5 tarts per sheet.

Sprinkle each with about 1.5 Tablespoons cheese. Spoon the mushroom topping, about a scant 1/4 cup for each tart, over the cheese. Press down lightly with the back of a spoon.

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

Remove from the oven and garnish with flaky sea salt, more herbs, cheese, and/or a drizzle of balsamic glaze if desired. 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.