1. Make polenta: Line a baking pan with plastic wrap and set aside. Melt butter in a large, heavy saucepan over medium heat and sauté onion until golden -- about 10 minutes. Stir in polenta, broth, 1/2 teaspoon salt, 1/2 teaspoon black pepper, and cayenne pepper and bring mixture to boil. Lower heat and cook until polenta thickens -- 10 to 15 minutes. Stir in 1 cup water, cheese, and chopped rosemary. Transfer polenta to prepared pan, spread evenly, and cool completely.
2. Grill polenta: Heat a grill until very hot. Invert polenta pan onto clean surface. Cut polenta into 3-inch triangles, brush each with oil, and place on grill. Cook until polenta is heated through and golden -- about 10 minutes. Place grilled polenta triangles on a platter and keep warm.
3. Grill vegetables: Place vegetables in a large bowl, add remaining olive oil, salt, and pepper, and toss to coat. Grill vegetables until they just begin to soften. Place grilled vegetables over polenta and serve immediately.