To make the dough: Combine the flour, sugar, and yeast in the bowl of a food processor. Next, add the butter and 4 tsp olive oil and process until uniform, about 30 seconds. With the food processor running, add the water in a steady stream, processing until the dough is mixed and no dry flour remains, about 15 seconds. Cover the processor bowl and let the dough sit for 15 minutes.
Add the salt and process until a consistent ball forms, about 30 seconds.
Knead the dough on a lightly oiled surface. Knead until the dough is smooth and supple, about 30 seconds. Place the dough in a lightly oiled bowl and rest for 90 minutes.
Transfer the dough onto a lightly floured surface. Divide the dough ball in half (about 260g per piece) and shape each into a tight, smooth ball. Any seams, pinch closed. Transfer each ball to a small bowl that has been lightly oiled. Turn the ball to coat in oil, placing seam side down. Cover each bowl with a tight-fitting lid. If you do not have a tight-fitting lid, cover each ball with a saran wrap and then place a dish towel over that. Refrigerate between 18 and 48 hours.
To Make the Sauce: Combine all sauce ingredients in the food processor bowl or blender. Pulse until all ingredients are combined and smooth. Transfer to a container with a lid and refrigerate until needed. You can make the sauce up to 4 days ahead.
For assembly: 4 teaspoons refined coconut oil or vegetable shortening, divided into two portions.
1 teaspoon olive oil, divided in half.
To Assemble: About 2 ½ hours before baking, remove your dough from the refrigerator and allow it to rest at room temperature for 30 to 45 minutes.
Grease the bottom and sides of two 12" round pizza pans or two 9" x 13" cake pans with 2 teaspoons of the refined coconut oil or shortening (using 4 teaspoons total). Place one dough ball in the center of each pan and drizzle each with ½ teaspoon of olive oil.
Using your hands, gently press the dough into the pan as evenly as possible. If the dough tightens up, don't force it. Cover the pan and let it rest for about 10 minutes before trying to press again. Once the dough is pressed into the pan, cover the pan loosely and let sit until puffy, 60 to 90 minutes.
Place your pizza stone on the middle rack of your oven for at least 45 minutes before baking. Preheat the oven to 500 F or 550 F, the highest your oven will allow.
Combine the cheddar and mozzarella cheese in a bowl.
Gently stretch the dough to cover the bottom of the pan completely (if it has sprung back from doing it previously. If it is already to the edges, there is no need to stretch again).
Top one dough with ½ of the sauce, spreading it evenly and ensuring some flows over the edge.
Top the sauce with ½ of the cheese mixture, covering the sauce entirely so that the cheese is spread to the edges.
Sprinkle ½ the Parmesan over the top.
Drizzle with 1 teaspoon of olive oil.
Repeat steps 6-10 with the second dough.
Bake the pizza in the pan directly on the pizza stone for 8-10 minutes, until the cheese is bubbly and browned. You want the rim of the pizza to be well-browned.
Remove the pizza when it is browned and crisp on the edges, carefully slide it out of the pan, and place it directly on the pizza stone. It may need help coming out of the pan; a small spatula can help pry it out.
Leave directly on the pizza stone for 2 to 4 minutes longer.
Transfer the pizza to a cooling rack and sit for 5 minutes before cutting, serving, and enjoying.