1. Make the polenta:Line a baking pan with plastic wrap and set aside. Melt the butter in a large, heavy saucepan over medium heat and sauté the onion until golden -- about 10 minutes. Stir in polenta, broth, 1/2 teaspoon salt, 1/2 teaspoon black pepper, and cayenne pepper and bring the mixture to a 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 the polenta:Heat a grill until very hot. Invert the polenta pan onto a clean surface. Cut the 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 the grilled polenta triangles on a platter and keep warm.
3. Grill the vegetables:Place the vegetables in a large bowl, add the remaining olive oil, salt, and pepper, and toss to coat. Grill the vegetables until they just begin to soften. Place the grilled vegetables over the polenta and serve immediately.