Put an inverted baking sheet or pizza stone on the bottom oven rack; preheat to 475 degrees F. Brush the bottom and side of a 10-inch deep-dish pizza pan or a 9-inch-round cake pan (preferably dark metal) with 2 tablespoons olive oil. Press the dough into the pan.
Combine the tomato sauce, garlic, oregano, 1/2 teaspoon salt and a few grinds of pepper in a bowl. Spread 1 cup of the sauce over the dough; scatter the onion on top. Transfer the pan to the hot baking sheet; bake until the crust puffs up slightly and starts browning, about 10 minutes.
Remove the pizza from the oven; top with the cheese, pepperoncini and pepperoni, then top with the remaining sauce. Return to the hot baking sheet and bake until golden and crisp around the edge, about 12 minutes. Let cool 5 minutes. Top with the basil before slicing.
Toss the greens with the remaining 2 tablespoons olive oil and the pepperoncini brine; season with salt and pepper. Serve with the pizza.