Combine white sugar and warm water in a large bowl or in a work bowl of a stand mixer. Sprinkle yeast over warm sugar water, and let stand for 5 minutes until yeast softens and forms a creamy foam. Stir 1 tablespoon olive oil into yeast mixture.

Stir 1/2 teaspoon salt into flour. Mix half of flour mixture into yeast water, and stir until no dry spots remain. Stir in remaining flour, a 1/2 cup at a time, mixing well after each addition. When dough has pulled together, turn it out onto a lightly floured surface and knead until smooth and elastic, about 8 minutes. You can also mix with a dough hook in a stand mixer.

Lightly oil a large bowl, place dough in the bowl and turn to coat with oil. Cover with a light cloth, and let rise in a warm place until doubled in volume, about 1 hour.

Combine crushed tomatoes, brown sugar, garlic powder, 1 teaspoon olive oil, and salt in a small saucepan. Cover the pan, and cook over low heat until tomatoes start to break down, about 30 minutes.

Preheat the oven to 450 degrees F (230 degrees C). Deflate dough and turn it out onto a lightly floured surface. Cut the dough into 2 equal pieces. Roll one piece into a 12-inch thin circle. Roll other half into a thicker, 9-inch circle.

Place 12-inch dough round into an ungreased 9-inch springform pan. Sprinkle dough with 1 cup of cheese. Shape sausage into a 9-inch patty and place in pan on top of cheese. Layer pepperoni, mushrooms, green pepper, red pepper, and remaining cheese on sausage patty. Top with 9-inch dough round and pinch edges to seal. Cut several 1/2-inch vent holes in top crust. Spread sauce evenly on the top crust, leaving a 1/2-inch border at the edges.

Bake pizza in the oven until the crust is set, cheese is melted, and sausage is cooked through, 40 to 45 minutes. Let hot pizza rest for 15 minutes before cutting into wedges and serving.