In a medium saucepan, bring 4 cups water to a boil. Whisking constantly, gradually add 3/4 cup yellow cornmeal. Reduce heat to medium-low; simmer, whisking frequently, until thick and creamy, 2 to 5 minutes. Remove from heat. Stir in 1/2 cup grated Parmesan. Season with salt and pepper.
In a small saucepan, heat 2 tablespoons over medium. Add 1/2 to 1 teaspoon red-pepper flakes. Cook, stirring occasionally, until red-pepper flakes begin to brown, 1 to 2 minutes. Serve polenta topped with chile oil.