Heat a grill to medium heat and rub the grate with a towel dipped in oil. In a large bowl, add the oil and the vegetables and toss. Season with salt and pepper, to taste.
Grill the vegetables, turning rarely, over medium heat until soft and charred, about 5 minutes for the asparagus, 10 minutes for the onions, 15 minutes for the peppers and mushrooms. Take vegetables off the grill and put in a bowl. Add the pesto sauce to a small bowl and add the grilled mushrooms. Toss to coat the mushrooms with the pesto.
For each burger, spread 1 tablespoon of the mustard on the bottom of the bun, top with 1/4 of the goat cheese, then layer on the mushrooms, onion, bell pepper, and asparagus. Cover with the top of the bun and serve.
You can also use an indoor grill to grill the vegetables.