Make dough: In the bowl of a standing mixer fitted with a dough hook, combine flour, water, and salt. Mix on low until evenly combined, 3 to 4 minutes.
Let dough rest 10 minutes.
Add yeast and sugar and mix for 2 minutes to evenly combine. Increase speed to high and mix until dough pulls away from the sides of the bowl, 4 to 6 minutes more, scraping down the sides of the bowl as needed. (You may be tempted to add more flour to help the dough come together, but don’t do it. This should be a very wet, floppy dough.)
Prepare two 8-inch square cake pans (see note) by spraying them with nonstick cooking spray and then generously rubbing them each with about 1 Tablespoon of olive oil. (Don’t skip these steps - you’ll want to do everything you can to help minimize sticking.)
Divide dough evenly between pans (it may not reach the edges). Drizzle each pizza with another 1 Tablespoon of olive oil.
Let rise, uncovered, at room temperature until dough reaches the sides of the pan and bubbles form on the surface, about 2 hours.
Make sauce: Heat 2 Tablespoons of olive oil in a small saucepan over medium heat. Add shallot and sauté until soft, about 2 minutes. Add red wine vinegar and cook until vinegar reduces, 1 to 2 minutes more. Stir in tomato paste, crushed tomatoes, sugar, and black pepper. Bring to a simmer, stirring frequently. Cook until thick, 2 to 3 minutes more. (If the sauce begins to sputter, reduce heat and add a lid, leaving it slightly ajar.) Set aside to cool.
Combine shredded mozzarella and cheddar cheeses. Refrigerate until ready to use.
Heat oven to 425°F / 220°C.
Assemble pizzas: When pizzas are done rising, scatter half the cheese mixture over top of the pizzas, all the way out to the edges (the cheese around the edges becomes the cheesy crust).
Spread thin lines of sauce over the tops of the pizzas (the sauce shouldn’t fully cover the top, but you want to be sure there’s some sauce in every slice).
Top pizzas with pepperoni and parmesan cheese.
Bake pizzas until the cheese is deep golden brown around the edges and bubbling on top, 18 to 22 minutes. (This may vary slightly depending on the thickness of the pans.)
Remove pizzas from oven and immediately run a flat spatula around the edges to loosen the cheese from the sides (it’s very important to do this right away to prevent sticking).
Let the pizzas rest in the pan for a minute before using a large spatula to transfer them to a wire rack to continue cooling.
Top pizza with generous spoonfuls of ricotta and fresh basil.
Slice into squares and serve!