Do Ahead Tip: Marinate for 30 minutes, (see recipe).
In a large freezer bag, add the first 6 ingredients (lemon juice through red pepper flakes) and 2 tablespoons oil. Add the chicken to bag, seal, and refrigerate for at least 30 minutes, or overnight. Soak wooden skewers for at least 15 minutes.
Thread wooden skewers with alternating pieces of chicken, onion, and mushroom. Heat a large grill pan over medium-high heat and add olive oil. Add skewers to the pan. Meanwhile, toss broccoli with remaining two tablespoons of avocado oil, balsamic vinegar, and salt and pepper to taste. Add broccoli to grill pan.
Cook chicken skewers and broccoli in the grill pan for 6-8 minutes until chicken is cooked through and broccoli is tender, turning occasionally to ensure even cooking. Serve with cauliflower and butter.