Add the flour, baking powder, salt, and sugar to a medium-sized mixing bowl and whisk with a fork.
Add the beer and olive oil to the flour mixture and stir with a fork until combined and the dough leaves the sides of a bowl.
Optional - the dough can also be made in a small food processor. Add the dry ingredients and pulse for 30 seconds then add the beer and olive oil to the food processor bowl and process just until the dough leaves the sides of the bowl.
Transfer the dough to a floured surface and knead the dough until it becomes a smooth ball.
Cover the dough ball with plastic wrap and let sit to relax the dough for 10 minutes.
Roll the dough into a 9" circle with a rolling pin, or you can stretch it with your hands.
Heat 10" skillet over medium heat, and add 1 tablespoon of olive oil. Most any skillet type will work, I use an iron-skillet. When the oil is hot add the pizza dough to the skillet and poke holes all over with a fork to keep the dough from puffing up. Cook 3-4 minutes until the bottom of the dough is golden brown and crispy.
Flip the dough over and add the drained tomatoes, cheeses, and basil.
Reduce the heat to low and cover the skillet. Cook for about 5 - 7 minutes until the bottom crust is brown and crispy and the cheese has melted.
Transfer the pizza to a cutting board, using tongs, a spatula, or depending on the pan, the pizza can slide out. Cut the pizza into four quarters and serve.