Rinse the vegetables and remove all of the skins, ends and stems. Slice the vegetables fairly thin and place on a cookie sheet. Drizzle the vegetables with olive oil and sprinkle with your favorite herbs and seasonings (garlic powder, salt, pepper, basil, parsley, thyme, etc.). Heat an outdoor grill to medium-high heat. Place the large vegetable pieces onto the grill and let them cook on both sides for a few minutes, or until they get some nice grill lines. Once the vegetables are tender, remove from the heat and place on the same cookie sheet you started with. Toss them a little bit in the leftover oil on the pan and seasonings.
For the sandwich, spread a generous portion of pesto on both sides of the bread. Place the sliced and shredded cheese on the bottom bun. Layer the sandwich with grilled vegetables and put the lid on. Place the sandwich into a warm panini maker and heat until the cheese is melted. You can also do this on a flat griddle if you have one. Once the cheese is melted, remove and serve.