Fill the pellet hopper of your Z Grills wood pellet grill with pellets. Set temperature to the HIGH setting (450 °F), close lid, and preheat pellet grill for 10 to 15 minutes.
In a large bowl, stir the vinegar, shallot, mustard, honey, thyme, and salt together. Slowly whisk in olive oil until it is well combined with the rest ingredients. Reserve 1/4 cup vinaigrette.
Add eggplant, zucchini, yellow squash, and portobellos to remaining vinaigrette in bowl, and toss together. Place vegetables on the grill grate, close the lid, and grill until lightly charred, 4 to 5 minutes per side. Then remove from grill, and set aside.
Toss together tomatoes and reserved 1/4 cup vinaigrette in a large cast-iron or other heatproof skillets. Place the skillet on the grill grate, close the lid, and grill until skins burst and juices begin to release, 12 to 15 minutes.
Place eggplant mixture on a large platter. Top with tomatoes and juices from skillet. Tear the burrata into pieces, and scatter over vegetables.
Top with basil leaves just before serving.