Place a grill pan over medium-high heat or prepare the barbecue (medium-high heat). Brush the vegetables with 1/4 cup of the oil to coat lightly. Sprinkle the vegetables with salt and pepper. Working in batches, grill the vegetables until tender and lightly charred all over, about 8 to 10 minutes for the bell peppers; 7 minutes for the yellow squash, zucchini, eggplant, and mushrooms; 4 minutes for the asparagus and green onions. Arrange the vegetables on a platter. The key to getting those great grill marks is to not shift the vegetables too frequently once they've been placed on the hot grill.
Meanwhile, whisk the remaining 2 tablespoons of oil, balsamic vinegar, garlic, parsley, basil, and rosemary in a small bowl to blend. Add salt and pepper to taste. Drizzle the herb mixture over the vegetables. Serve the vegetables, warm or at room temperature.