2. Add the chicken broth, bay leaf, a pinch of salt and a pinch of black pepper. Increase the heat and bring the broth to a boil.
3. Slowly whisk the cornmeal into the hot broth. Cook for 15 to 20 minutes, stirring occasionally, until the polenta has thickened.
4. Remove the pan from the heat, stir in the basil and season with salt and pepper. (The polenta can be made in advance and re-warmed before serving.)
For the grilled vegetables:
1. Preheat the grill.
2. In a food processor, combine the garlic, mustard and vinegar and puree. With the motor running, add the olive oil and chicken stock. Season with salt and pepper. Brush the vegetables with the garlic vinaigrette.
3. Grill the vegetables on a hot grill for 2 minutes on each side. (To keep the onions together, slide a spatula completely underneath each slice to pick up all the rings and turn carefully to grill the other side.) Season to taste with salt and pepper.
4. Spoon the warm polenta onto serving plates and arrange the slices of grilled vegetables and mushrooms on top. Sprinkle with Parmesan cheese, garnish with basil leaves and serve.