Dissolve sugar in warm water in the bowl of a stand mixer fitted with a dough hook. Sprinkle yeast over water, and let stand for 5 minutes until yeast softens and begins to form a creamy foam.

Stir in oil, salt, and 2 cups of flour; 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 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; place dough in the bowl and turn to coat with oil. Cover with a towel and let rise in a warm place until doubled in volume, about 1 hour.

Combine ricotta, mozzarella, Parmesan cheese, pepperoni, parsley, Italian seasoning, olives, green pepper, and mushrooms in a large bowl; set aside.

Preheat the oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.

Punch down dough; roll it out into a long, flat strip about 9 inches wide by 30 inches long. Spoon filling mixture down the center of dough strip, leaving about 1 inch on all sides for sealing. Pull edges of dough together; pinch to seal and form a long, filled roll.

Place roll, seam-side down, onto the parchment paper in a snakey "S" shape, and tuck the ends underneath to seal. Beat egg in a bowl with 1 tablespoon of water; brush egg wash over calzone.