Heat oil over medium heat in a 4-qt. pot. Cook beef until browned, 3 to 4 minutes. Add onion, garlic, 1/4 cup basil, the oregano, rosemary, chile flakes, 1/2 tsp. salt, and the pepper and cook until onion is softened, 5 to 6 minutes. Add tomato paste and diced tomatoes and simmer until thickened, about 15 minutes.
Meanwhile, bring 1 qt. water to a boil in a medium pot. Add polenta and remaining 1/2 tsp. salt and cook, stirring, until polenta thickens, 3 to 4 minutes. Stir in butter and 1/3 cup cheese. Serve polenta topped with ragù, basil, and more cheese.