This is a centerpiece fit for a celebratory gathering. We added extra pine nuts and reduced the olive oil a bit to make a chunky pesto that spreads easily but does not melt out in the oven. Any mozzarella bits sticking out of the "branches" become golden brown and crunchy.
Position an oven rack in the lower third of the oven and preheat to 375 degrees F. Line an upside-down baking sheet with parchment and set aside.
Add the basil, pine nuts, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper to a food processor and pulse until coarsely chopped. With the motor running, drizzle in 1/4 cup of the olive oil until incorporated. The pesto should be slightly chunky, not smooth.
Combine the mozzarella, ricotta, pecorino, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl. Add the pesto to the cheese mixture and stir to combine.
Roll out the pizza dough into a 12-by-15-inch rectangle. Using a pizza cutter, cut the dough into 1 large and 2 small triangles; one of the 12-inch sides will be the base of the large triangle with the 2 small triangles on either side. Combine the 15-inch sides of the 2 small triangles to create a large triangle on the prepared baking sheet, tugging and pressing down on the dough as needed to shape it. This will be the base of the tree.
Leaving a 1/4-inch border around the perimeter of the triangle, spread the pesto and cheese mixture evenly to cover the base. Cover it with the remaining large triangle, tugging the dough as needed to fit over the base. Seal the 1/4-inch border by firmly pressing the top edge down onto the bottom.
Leaving a 1-inch "trunk" down the middle of the tree intact, use a sharp knife to cut horizontal lines, about 1 1/2 inches apart, from the middle "trunk" to each edge, making "branches." Twist each "branch" to reveal the layers of cheese and pesto. The shorter ones will twist once or twice. The longer ones will twist three or four times.
Bake until golden brown, 20 to 25 minutes. Lightly brush with the remaining 1 teaspoon olive oil. Serve warm.
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