Make the marinade. In a bowl, whisk balsamic vinegar, olive oil, minced garlic, honey, Dijon, Italian seasoning, salt, and pepper.
Taste and adjust sweetness or acidity if needed.
Prep the chicken. Pat the chicken dry. Place in a zip-top bag or shallow dish. Pour in the marinade, turning to coat.
Marinate for at least 30 minutes, or up to 8 hours in the fridge.
Prep the veggies. In a large bowl, toss peppers, red onion, zucchini, yellow squash, cherry tomatoes, and broccoli with olive oil, balsamic, Italian seasoning, salt, and pepper. Set aside.
Heat the grill or oven. Preheat a grill to medium-high (about 400–425°F). If cooking indoors, preheat the oven to 425°F and line a sheet pan with parchment.
Grill the chicken. Remove chicken from marinade, letting excess drip off.
Grill 5–6 minutes per side for breasts (thighs may take a bit longer), until the internal temperature reaches 165°F. Brush with leftover marinade during the first half of cooking only, then discard the rest.
Cook the veggies. Grill in a grill basket for 8–12 minutes, tossing occasionally, until charred in spots and tender-crisp. If roasting, spread on the sheet pan and roast for 18–22 minutes, stirring once, until caramelized.
Rest and slice. Let the chicken rest 5 minutes, then slice against the grain.
This keeps it juicy and easier to serve.
Finish and serve. Toss the veggies with chopped basil or parsley. Plate with chicken, add shaved Parmesan, and squeeze lemon over the top for brightness.