On a lightly floured surface, form each piece of cold dough into a rounded mound. Cover with a kitchen towel and let come to room temperature, about 20 minutes. Meanwhile, in a large skillet, heat 1/4 cup EVOO over high heat. Add the mushrooms, season with salt and pepper and cook, stirring occasionally, until browned, about 7 minutes.
Position a rack in the center of the oven and preheat to 400 degrees . Line a large baking sheet with parchment. Roll out a dough mound and stretch into a 7-by-8-inch rectangle. With a long end facing you, arrange one-quarter of the pepperoni in a overlapping line across the center of the dough, leaving a 1-inch border at each short end. layer with one-quarter of the mozzarella, then one-quarter of the mushrooms. Fold 2 slices swiss in half and lay them side by side on top of the mushroom layer. Finally, top with one-quarter of the turkey. Starting with the side farthest from you, stretch the dough to cover the filling completely, being careful not to tear it. Repeat with the side nearest to you, brining it in to overlap the other piece of dough. press the edges all the way around the hero to seal. Transfer the hero seam side down to the baking sheet. Repeat with the remaining ingredients to form 3 more heros.
Brush the outsides of the dough with a little EVOO and bake until browned, about 25 minutes. let cook for 20 to 30 minutes. Slice each hero in half crosswise, then serve with the mustard or marinara sauce for dipping.