In a medium bowl, toss together the tomatoes, garlic and salt. Transfer to a colander and drain for 15 min. Transfer tomatoes to a paper towel-lined plate to remove excess moisture. In another medium bowl, combine the cheeses and basil; refrigerate while preparing the dough.

In the bowl of a food processor, combine the flour, baking powder, sugar, and salt. With the motor running, slowly add the beer and 1 tablespoon of the oil and process until the dough pulls away from the sides and forms a shaggy ball, about 1 minute. Using floured hands, form dough into a tight ball and cover loosely with plastic wrap; let rest 10 minutes. (Dough can be wrapped tightly in plastic wrap for up to 1 week.)

Divide dough in half. On a lightly floured surface, roll each half into a very thin 9-inch circle. Heat 3 tablespoons of oil in a 12-inch nonstick skillet over medium heat until just smoking. Transfer 1 dough round to skillet and cook. Using the tines of a fork, gently poking any bubbles that form. Cook until the bottom is deep golden brown and crisp, about 3-4 minutes. Flip dough and sprinkle with half of drained tomatoes and and half of cheese mixture. Reduce heat to low and cook, covered, until the second side is crisp and the cheeses have melted, about 5 minutes. Transfer pizza to a wire rack. Wipe out pan and repeat with remains oil, dough, and toppings. Transfer pizzas to a cutting board; slice and serve. Enjoy!