Preheat oven to 450 degrees F. Bring a large pot of salted water to a boil.
Heat the oil in a medium saucepan over medium heat. Add the garlic and red pepper flakes and cook, stirring, until the garlic is golden, about 2 minutes. Add both cans of the tomatoes and 1 cup of water. Increase the heat to medium-high and bring to a simmer. Reduce the heat slightly and simmer rapidly, uncovered, until thick, 15 to 20 minutes. Stir in the ricotta and season with salt and pepper.
While the sauce simmers, put the bacon in a large skillet and cook over medium heat, stirring often, until crispy, about 10 minutes. Transfer to a large bowl with a slotted spoon. Discard all but about 2 tablespoons of the bacon drippings from the skillet then add the mushrooms and cook, stirring, until golden brown, about 8 minutes. Add the mushrooms to the bowl with the bacon.
Cook the farfalle in the boiling water until very al dente, about 2 minutes less than the package directions. Drain the pasta and add it to the bowl.
Add the tomato sauce, the Asiago and half of the pecorino to the bowl and stir well to combine. Transfer the pasta to a 3- to 4-quart baking dish and top with the mozzarella and the remaining pecorino. Bake, uncovered, until browned, about 15 minutes. Let the pasta rest for about 10 minutes before serving.
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