Preheat the oven to 400°F (200°C).
In a 9x13-inch baking dish, combine the cherry tomatoes, 1/4 cup of the olive oil, and a pinch of salt and pepper. Toss to coat the tomatoes thoroughly.
Place the block of feta cheese in the center of the baking dish. Drizzle the remaining 1/4 cup of olive oil over the feta and sprinkle with red pepper flakes and dried oregano.
Bake for 30 minutes. Increase the oven temperature to 450°F (230°C) and bake for an additional 5 to 10 minutes, or until the tomatoes have burst and the feta is slightly golden on top.
While the feta is baking, cook the pasta in a large pot of salted boiling water according to package instructions until al dente. Reserve 1/2 cup of the pasta water before draining.
Remove the baking dish from the oven and immediately add the minced garlic. Use a fork to mash the softened feta and burst tomatoes together until a creamy, uniform sauce forms.
Add the cooked pasta and torn basil to the baking dish. Toss well to coat the pasta in the sauce, adding small amounts of the reserved pasta water if the sauce is too thick.
Season with additional salt and pepper to taste and serve immediately.