Up to 2 days ahead, make the mushroom topping. Place a large sauté pan over medium-high heat and drizzle in the olive oil. Add the onions and stir until they start to sizzle. Reduce the heat to medium-low and cook, stirring occasionally, for at least 10 minutes, and up to an hour to caramelize.
Add the shiitake mushrooms and garlic and increase the heat to medium-high again. Add the sage, salt and pepper, and stir until the mushrooms soften and shrink. Reduce the heat and stir for about 5 minutes to shrink the mushrooms. When soft, transfer to a medium bowl and let cool completely. Stir in the cheese and reserve. Can be kept in the refrigerator, tightly covered, until ready to bake.
To bake, preheat the oven to 400 degrees. Line a baking sheet with parchment paper and reserve.
On a lightly floured counter, unfold the puff pastry and use a rolling pin to roll a few times to smooth out the fold marks and make the dough into a 10- by 13-inch rectangle, then place the dough on the prepared pan. Whisk the egg, if using, with a teaspoon of water and brush a 1-inch border around the puff pastry rectangle. Use a fork to prick the dough every 2 inches inside the border.
Sprinkle the mushroom mixture evenly over the dough, leaving a 1/2-inch border bare. Sprinkle with roasted pepper strips over the mushrooms.
Bake for about 15 minutes, until the edges of the puff and the cheese are both golden brown.