Dissolve the sugar in the warm water in the bowl of a stand mixer fitted with a dough hook. Sprinkle the yeast over the water, and let stand for 5 minutes until the yeast softens and begins to form a creamy foam. Stir in the oil, salt, and 2 cups of flour, and mix on Low until the mixture forms a wet dough. With the machine running, add 1 cup of flour, 1/4 cup at a time, and knead until the dough pulls away from the sides of the bowl into a ball. Knead for 3 to 5 more minutes, until the dough is smooth and elastic.
Lightly oil a large bowl, then place the dough in the bowl and turn to coat with oil. Cover with a light cloth and let rise in a warm place (80 to 95 degrees F (27 to 35 degrees C)) until doubled in volume, about 1 hour.
Combine the ricotta, mozzarella, and Parmesan cheeses, pepperoni, parsley, Italian seasoning, olives, green pepper, and mushrooms in a large bowl, and set aside.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Punch down the dough, and roll it out into a long, flat strip about 9 inches wide by 30 inches long. Spoon the filling mixture down the center of the dough strip, leaving about 1 inch on all sides for sealing. Pull edges of the dough together; pinch to seal and form a long, filled roll. Place the roll, seam side down, onto the parchment paper in a snakey "S" shape, and tuck the ends underneath to seal. Beat the egg in a bowl with 1 tablespoon of water, and brush the egg wash over the calzone.
Bake in the preheated oven for 30 to 35 minutes, until the calzone is golden brown. Let cool for 5 minutes before cutting into individual slices.