Baked Broccoli Polenta with Roasted Mushrooms

Name: Baked Broccoli Polenta with Roasted Mushrooms
Ingredients: ¼ cups plus 2 tablespoons extra-virgin olive oil, divided; 1 small head of broccoli, stems peeled, coarsely chopped into ¾-inch chunks (2 cups); 1 tsp. kosher salt, divided; Pinch crushed red chile flakes; 2 garlic cloves, coarsely chopped; 3½ cups chicken or vegetable broth; 2 tsp. unsalted butter; 1 cup medium-ground polenta; ½ cups (1¾ oz.) finely grated Pecorino Romano; 1 lb. maitake mushrooms, gently torn in bite-size pieces; ¼ cups extra-virgin olive oil; ½ tsp. fine sea salt; Flaky sea salt and cracked black pepper