In a bowl, whisk together balsamic vinegar, olive oil, minced garlic, and thyme to create the marinade.
Place mushroom caps in a shallow dish and brush both sides with the marinade. Let them rest for 15-20 minutes.
Meanwhile, brush zucchini slices with remaining marinade and season with salt and pepper.
Preheat your grill to medium-high heat (around 400°F).
Grill the mushrooms gill-side up for 5-6 minutes, then flip and cook for another 3-4 minutes. If using cheese, add it during the last minute of cooking.
Grill zucchini slices for 2-3 minutes per side until tender with nice grill marks.
Grill onion rings until slightly charred and softened, about 3-4 minutes per side.
Lightly toast the burger buns on the grill for 30 seconds.
Assemble burgers by layering grilled zucchini, mushroom caps, onions, and fresh basil on the toasted buns.