Line a baking pan with plastic wrap and set aside for the polenta.

Melt butter in a large saucepan over medium heat.

Sauté the onion in the melted butter until golden, about 10 minutes.

Stir in the polenta, chicken broth, 1/2 tsp salt, 1/2 tsp black pepper, and cayenne pepper.

Bring the mixture to a boil, then reduce the heat to low.

Cook until the polenta thickens, stirring occasionally, about 10-15 minutes.

Stir in 1 cup of water, goat cheese, and chopped rosemary.

Transfer the polenta to the prepared baking pan and spread evenly.

Cool the polenta completely until firm.

Preheat a grill until very hot.

Invert the polenta pan onto a clean surface.

Cut the polenta into 3-inch triangles.

Brush each polenta triangle with olive oil.

Place the polenta triangles on the grill and cook until heated through and golden, about 10 minutes.

Place the grilled polenta triangles on a platter and keep warm.

Place the mixed vegetables in a large bowl.

Add the remaining olive oil, salt, and pepper to the vegetables.

Toss to coat the vegetables evenly.

Grill the vegetables until they just begin to soften.

Place the grilled vegetables over the polenta and serve immediately.