Preheat the Oven: If your broiler is in the oven, preheat to 500°F. If in a separate drawer, preheat the oven to 450°F. Place a pizza stone or rimless baking sheet in the oven and allow it to heat for 45 minutes.
Prepare the Dough: Allow the pizza dough to sit at room temperature for 30 minutes.
Cook the Anchovies: Heat a small skillet over high heat. Add ¼ cup olive oil and anchovy fillets, mashing the anchovies with a spoon until dissolved (about 2 minutes). Set aside.
Shape the Dough: Dust a rimmed baking sheet (or pizza peel) generously with cornmeal. Shape and stretch one ball of dough into a 12-inch round directly on the surface. Scatter half of the tomatoes, garlic, and oregano over the dough. Top with half of the cheese, drizzle with half of the anchovy oil, and sprinkle with salt and pepper to taste. Ensure the pizza is not sticking to the baking sheet or peel; add more cornmeal as needed.
Cook the Pizza: Slide the prepared pizza onto the preheated pizza stone or baking sheet in the oven. Cook for 3 minutes. Then broil until golden, crisp, and blistered in places (2–4 minutes). For an oven without a broiler, bake at 450°F for 10–20 minutes.
Finish and Serve: Using tongs, transfer the cooked pizza to a serving platter. Drizzle with more olive oil, garnish with fresh basil leaves, and repeat the process with the remaining dough and toppings.