Preheat grill to medium-high heat.
Place eggplants, onion, and cherry tomatoes in separate aluminum tins. Toss each veggie with some of the olive oil and season with salt, pepper, and lemon pepper. Set aside.
Grill each of the veggies until the eggplant is nicely browned and has some crispy edges, but still in tact, the tomatoes are blistered and slightly blackened, and the onions are caramelized. The eggplant will cook the fastest (about 7 minutes depending upon the heat of your grill),
The tomatoes will take bout 10 minutes or so.
And, finally the onions may take up to 15 minutes or so. Stir each occasionally as they are cooking. This can all be done several hours ahead of time and refrigerated until ready to serve.
Meanwhile, grate the tomatoes on a cheese grater and add to a medium-sized pot. Add the minced garlic and basil. Heat over low heat for about 5 minutes. Set aside until veggies are redy and the pasta is cooked.
To assemble this dish, add the grilled veggies to the tomato sauce and heat until hot. Keep it hot until the pasta is cooked. When the pasta is al dente, drain it and toss it into the sauce. Serve on heated plates and garnish with a scoop of burrata, a generous sprinkling of Parmesan, and a basil leaf or minced basil.