In an 8 by 8-inch glass baking dish, mix together the mustard, lemon juice, Worcestershire sauce and garlic. Season with fresh ground pepper to taste.
Pound the chicken to an even thickness, about 1/2-inch thick. Butterfly larger pieces and cut in half. Roll each piece in the mustard mixture, coating both sides. Let marinate in the dish with the sauce while you make the caramelized onions.
In a large heavy saucepan, melt the butter and olive oil over medium heat.
Stir in the sugar and then add the sliced onions and cook just until they start to sizzle, then turn down to medium-low and cook for 15 to 20 minutes or until the onions are soft and starting to brown. Stir frequently so they don’t stick to the pan.
Drain off any excess oil or butter (you may not have any), add the wine and balsamic vinegar. Bring to a boil, then turn heat down immediately to medium-low and cook until the liquid has been absorbed, around 10 to 15 minutes. Season to taste with salt and pepper. Set aside to cool.
Place chicken breasts on grill over medium heat. Grill for around 3 to 4 minutes, turn over and grill for an additional 2 to 3 minutes or until an instant-read thermometer inserted into thickest portion registers 165 degrees (they cook fast since you have pounded them down a bit).
Place on dinner plate or serving platter with caramelized red onions on top.
MAKE AHEAD: Onions can be prepared up to 3 weeks ahead, covered and refrigerated. Chicken breasts can be marinated for up to 8 hours.
GLUTEN FREE: Use gluten free Worcestershire sauce.