In the bowl of a stand mixer fitted with a dough hook mix flour, salt, sugar, and yeast on low speed until incorporated, approximately 1 minute. To your dry ingredients add water and melted butter and mix on low speed until fully combined, 1 to 2 minutes, scraping sides and bottom of bowl occasionally. Increase speed to medium and knead until dough is glossy and smooth and pulls away from sides of bowl, 4 to 5 minutes. (Dough will only pull away from sides while mixer is on. When mixer is off, dough will fall back to sides.)
Coat a large bowl with 1 teaspoon olive oil. Transfer dough to bowl, turning once to oil top; cover tightly with plastic wrap. Let rise at room temperature until nearly doubled in volume, 45 to 60 minutes.
FOR THE SAUCE
In medium saucepan over medium heat melt butter. Add grated onion, oregano, and 1/2 teaspoon salt; cook, stirring occasionally, until liquid has evaporated and onion is golden brown, approximately 5 minutes. Add garlic and cook until fragrant, about 30 seconds. Stir in tomatoes and sugar, increase heat to high, and bring to simmer. Lower heat to medium-low and simmer until reduced to 2 1/2 cups, 25 to 30 minutes. Off heat, stir in basil and oil, then season with salt and pepper.
TO LAMINATE THE DOUGH
Adjust oven rack to lower position and heat oven to 425 degrees. Using rubber spatula, turn dough out onto dry work surface and roll into 15- by 12-inch rec-tangle. Using offset spatula, spread softened butter over surface of dough, leaving 1/2-inch border along edges. Starting at short end, roll dough into tight cylinder. With seam side down, flatten cylinder into 18- by 4-inch rectangle. Cut rectangle in half crosswise. Working with 1 half, fold into thirds like business letter; pinch seams together to form ball. Repeat with remaining half. Return balls to oiled bowl, cover tightly with plastic wrap, and let rise in refrigerator until nearly doubled in volume, 40 to 50 minutes.
Coat two 9-inch round cake pans with 2 tablespoons olive oil each. Transfer 1 dough ball to dry work surface and roll out into 13-inch disk about 1/4 inch thick. Transfer dough to pan by rolling dough loosely around rolling pin and unrolling into pan. Lightly press dough into pan, working into corners and 1 inch up sides. If dough resists stretching, let it relax 5 minutes before trying again. Repeat with remaining dough ball.
For each pizza, sprinkle 2 cups mozzarella evenly over surface of dough. Spread 1 1/4 cups tomato sauce over cheese and sprinkle 2 tablespoons Parmesan over sauce. Bake until crust is golden brown, 20 to 30 minutes. Remove pizza from oven and let rest 10 minutes before slicing and serving.