Melt the butter in a large skillet over medium heat, then add the shallots and let them soften for a few minutes. Stir in the mushrooms, garlic, salt, pepper, thyme, and rosemary, cooking until everything turns tender and fragrant. Remove the pan from the heat and stir in the balsamic vinegar right at the end, which wakes up the whole mixture.
On a lightly floured surface, roll the thawed or homemade dough into a large rectangle, roughly 10 by 17 inches. If you're using store-bought pastry sheets, overlap them slightly and roll them together so they behave like one piece. Use a pizza cutter to slice the dough into ten smaller rectangles, each about the size of your palm.
Brush every rectangle generously with the egg wash, then crimp the edges with a fork just like you would a pie crust. This creates a little raised border that keeps the filling from spilling over as it bakes. Sprinkle crumbled cheese over the center of each piece before adding your mushroom mixture.
Spoon a scant quarter cup of the cooled mushroom mixture over the cheese on each pastry rectangle, staying inside the crimped edge. Gently press down with the back of a spoon so the topping sticks in place. This little press keeps everything from sliding around once the pastry puffs up in the oven.
Preheat your oven to 400°F and arrange the tarts on lined baking sheets, leaving space between each one. Bake for 20 to 22 minutes, until the pastry turns deep golden brown and puffs up around the edges. You'll know they're ready the moment your kitchen starts smelling incredible.
Let the tarts rest for a couple of minutes once they come out of the oven, since the filling is very hot. Finish with a sprinkle of flaky sea salt, extra fresh herbs, or a light drizzle of balsamic glaze for a restaurant-style finish. Serve them warm or at room temperature, whichever you prefer.