Cut chicken into 1 1/2 - 2 inch cubes.
Cut broccoli into spears.
Mix marinade in a bowl: combine balsamic vinegar, lemon juice, olive oil. brown sugar, rosemary, garlic, thyme, basil, salt and pepper. Whisking until well combined. Set 1/3 cup of marinade aside for grilling.
Add chicken, stir to coat well. Cover and refrigerate one hour.
Prepare a grill fire to about 300° with hickory or pecan.
Thread chicken and broccoli onto skewers.
Spray or brush skewers with olive oil.
Place skewers over indirect fire.
Brush with reserved marinade.
Cook with the grill closed for about 15 minutes, brush with marinade once more. Cook another 10-15 minutes, check with digital thermometer, cooking until internal temperature reaches 165°.
Remove from grill and cover with foil, resting while you prepare the cheese sauce.