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To make the dough: In the bowl of a food processor, combine the flour, sugar, and yeast. Add the butter and oil and process until completely uniform, about 30 seconds. With the machine running, add the water in a steady stream. Continue to process until the dough is just mixed and no dry flour remains, about 10 to 15 seconds. Allow the dough to sit in the food processor, covered, for 15 minutes.
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Add the salt and process until an even ball forms and the sides of the bowl are clear, 15 to 30 seconds.
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On a lightly oiled work surface, knead the dough until smooth and supple, about 30 seconds. Place in a lightly oiled bowl, cover, and let rest until slightly puffy, about 90 minutes.
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Transfer the dough to a lightly floured work surface and divide in half (about 260g per piece) and shape each into a tight, smooth ball, pinching the seams on the bottom closed. Transfer each ball to a small, lightly oiled bowl, turn to coat in oil, then place seam-side down. Cover each bowl tightly and place in the refrigerator for at least 18 hours and up to 48 hours.
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To make the sauce: In the bowl of a food processor or blender, combine all the sauce ingredients and pulse until smooth. Transfer to a container and refrigerate until needed; it can be made several days ahead.
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To assemble and bake the bar pizzas: About 2 1/2 hours before baking, remove the dough from the refrigerator and allow it to rest at room temperature to warm up slightly, about 30 to 45 minutes.
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Grease the bottom and sides of two 12" round pizza pans or two 9" x 13" pans with 2 teaspoons of the coconut oil or shortening each (use 4 teaspoons total). Place one dough ball in the center of each pan and drizzle each lightly with 1/2 teaspoon of the olive oil.
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Using your hands, gently press the dough toward the sides of the pans as evenly as possible (the dough will slide across the grease into a thin layer). If the dough tightens up, don’t force it; cover the pan and let it sit for 5 to 10 minutes before pressing once again. Cover the dough in the pan loosely and let sit until puffy, 1 to 1 1/2 hours.
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Place a baking steel or stone on the middle rack of your oven about 45 minutes before baking, and preheat the oven to 500˚F to 550°F (or the highest temperature your pan allows).
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In a small bowl, combine the cheddar and mozzarella cheeses. Working with one pizza at a time, gently stretch the dough so that it covers the pan completely if it’s not already. Top the dough with 1/2 cup of the sauce, spreading it evenly to the edge of the pan and making sure some flows over the edge toward the bottom of the pan. Cover the sauce completely and evenly with half of the cheese mixture, making sure to cover the edges thoroughly. Sprinkle about 1/4 cup of the Parmesan over the pizza and drizzle with 1 teaspoon of the olive oil.
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Bake the bar pizza on the steel or stone for about 8 to 10 minutes, until the cheese is bubbly and browned and the rim of the pie is well-browned. While the first bar pizza bakes, top the second one.
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After 8 to 10 minutes, remove the first pizza from the oven, and use an offset spatula to loosen the rim of the pizza from the pan. Carefully slide the pizza out of the pan and place it directly on the steel or stone until browned and crisp, 2 to 4 minutes longer. Transfer the pizza to a wire rack and allow it to cool for 5 minutes before cutting and serving. While the first pizza cools, bake the second one.
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Storage information: Bar pizza is best eaten immediately, though leftovers can be stored in the refrigerator in an airtight container for several days. Reheat individual slices on the stovetop in a cast iron pan or in a toaster oven until warm.
