Whisk together pesto and olive oil in a large baking dish. Add vegetables and toss to coat. Let vegetables marinate for 3–4 hours.
Preheat oven to 400°F. Place vegetables on baking sheet and season with salt and pepper. Bake for 30–35 minutes, or until vegetables are tender and onions are beginning to brown.
Heat panini grill to medium-high. Top four slices of bread with 2 tablespoons of cheese and then divide roasted veggies evenly among the bread slices. Top each sandwich with another slice of bread and place sandwiches on panini grill. Grill until cheese has melted and bread is crispy, about 10–12 minutes.
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