Preheat grill on high heat. Prepare pasta according to package instructions and set aside.
Peel the red onion then slice into thick rounds. To keep the rounds intact while grilling, insert 2-3 toothpicks around the outside edge; set aside.
While the grill is heating and pasta is cooking, whisk together olive oil, garlic, oregano, salt and pepper in a large mixing bowl. Add the vegetables to the bowl (peppers, eggplant and onion) and toss well to coat.
Place vegetables on a single layer on the grill and cook until slightly charred and tender, about 5-6 minutes per side, flipping once. Keep the bowl and any reserved marinade next to your grill. When vegetables are done cooking, from the grill and place back in the bowl with the reserved oil mixture.
When vegetables are cool enough to touch, remove from bowl and roughly chop into bite-sized pieces. Place back in the bowl with the reserved oil mixture along with the cooked pasta. Add cheese and basil then toss again. Check for seasoning and add salt and pepper to taste. Serve warm, room temperature or chilled.