Bring a large pot of generously salted water to a boil, then add the spaghetti and cook, stirring frequently, until al dente, 8–10 minutes.
Meanwhile, to a large skillet over medium heat, add ¼ cup of the oil, the garlic, and chile flakes. Cook, stirring continuously, until the garlic is softened but not yet browned, about 2 minutes. Remove from the heat.
Using tongs, transfer the spaghetti to the skillet and turn the heat to low. Add ¼ cup of the pasta water and cook, tossing frequently, until the spaghetti is coated in a thin pan sauce, about 1 minute (add a few more tablespoons of pasta water if the spaghetti seems dry). Add half of the Pecorino and the remaining olive oil, tossing well.
Transfer the spaghetti to a large platter, or divide among wide pasta bowls, and top with the remaining Pecorino. Serve immediately, with additional Pecorino on the side if desired.