Combine flour, baking powder, sugar and salt in food processor. Add beer and 1 tablespoon olive oil and process until dough forms a shaggy ball, about 1 minute. Form dough into a ball, cover loosely with plastic wrap and let rest 10 minutes.
Divide dough in half. Roll each half into a very thin 9-inch round. Heat 3 tablespoons olive in a large non-stick skillet over medium heat until just smoking. Place one dough round into skillet and cook, poking any bubbles that form with fork, until bottom is golden brown and crisp, 3 to 4 minutes.
Mix the cheese and dried oregano together in a small bowl. Flip dough and top with tomato sauce and half of cheese mixture. Reduce heat to low and cook, covered until second side is crisp and cheese has melted, about 5 minutes. Broil for 2 to 3 minutes to get a nice crisp top crust.