Preheat your oven to 475 degrees F. Put a large pizza pan or cast-iron braising pan (which is what I used) in the oven while it heats up, so that it gets very hot.
While the oven is heating up, make the heirloom tomato mixture. Wash and dry the heirloom tomatoes, then cut them in half and add them to a mixing bowl. Add in the olive oil, lemon juice, salt, and pepper. Grate a clove of garlic into the mixture using a microplane, and stir until everything is evenly mixed. Set aside.
On a floured surface, shape your pizza crust until it is about 1/2" - 3/4" in height.
CAREFULLY take the prepared pan out of the oven, and place it on a heat-safe surface.
Carefully add in your pizza crust, then drizzle the olive oil over it. Spread the olive oil around so that the entire top of the crust is coated.
Add on the layer of shredded mozzarella or Italian blend cheese.
Using a slotted spoon, scatter the marinated tomatoes onto the pizza, trying to avoid the liquid left at the bottom of the boil. Then, break up your ball of burrata and add pieces of it among the tomatoes, as much or as little as you'd like.
Top with fresh oregano (save the basil for after cooking).
At the pizza pan back into the oven, and cook for about 22-26 minutes, or until the cheese is melted, and the crust is golden brown. Check often near the end of cooking to prevent burning.
Take the pizza out of the oven, and top with the fresh basil. Let it cool slightly, then carefully remove from the pan. Feel free to garnish with more black pepper or spicy chili flakes, if you'd like.
Cut into slices and enjoy!