Grilled Vegetables with Rosemary-Goat Cheese Polenta

• 1 Tbsp butter (unsalted)
• 1/2 cup onion, finely chopped 
• 3/4 cup polenta or stone-ground yellow cornmeal
• 1 can (14-ounce) reduced sodium chicken broth (or vegetable broth)
• 1 1/2 tsp coarse salt
• 1 1/2 tsp fresh-ground pepper
• 1/4 tsp cayenne pepper
• 3 Tbsp goat cheese
• 2 Tbsp rosemary, roughly chopped
• 1/2 cup extra-virgin olive oil
• 2 Tbsp extra-virgin olive oil
• 6 pound(s) mixed vegetables such as asparagus, cabbage, Japanese eggplant, summer squash, red potatoes, bell peppers, halved or quartered

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.