Place the tomatoes in a colander to drain for 30 minutes, tossing occasionally. Reserve the juice for later use, if desired.
☞ TESTER TIP: If your tomatoes are still quite wet after draining, lay them on paper towel to absorb excess moisture.
Line a 9-inch (23-cm) cast-iron skillet or deep-dish pie pan with the dough and prick the bottom of the crust several times with a fork to keep it from puffing up while baking. Line the dough with parchment paper. Place pie weights or rice over the bottom and up the sides of the paper.
Bake until the edges are beginning to brown, about 20 minutes. Remove the paper and weights and return to the oven to bake until the interior crust is light golden, about 10 minutes more. Place the pie crust in the skillet or pan on a rack to cool.
Meanwhile, in a medium bowl, combine the ricotta, eggs, Manchego, breadcrumbs, garlic, chives, parsley, 1 teaspoon of the black pepper, 1/2 teaspoon of the red pepper flakes, and 1 teaspoon of salt. Mix well.
Evenly spread the ricotta filling into the crust. Arrange the tomatoes over the cheese. Sprinkle the remaining pepper, red pepper flakes, and salt over the tomatoes. Drizzle with the olive oil.
Bake the pie until the tomatoes are browned, 45 to 60 minutes. Place the pie on a rack to cool for at least 20 minutes before slicing to serve.