Preheat the grill to high heat, approximately 400 – 450 degrees.
Pat the chicken dry with paper towels, then place them in a large ziplock plastic bag or between 2 pieces of plastic wrap. Use a plastic meat mallet or the palm of your hand to pound the chicken to an even thickness, about ½ inch.
Brush the chicken with the olive oil and sprinkle evenly with salt and pepper. Place on the preheated grill and cook, without turning, for 5 minutes. Turn the chicken and cook on the other side for another 4 to 5 minutes.
Remove the chicken to a platter and baste immediately with the vinaigrette. Loosely cover the chicken with aluminum foil and allow to rest for 5 minutes. Serve drizzled with more vinaigrette.
Preheat a small sauté pan — do not use cast iron — over medium high heat until hot, but not smoking.
Dip the cut side of each lemon half in the raw sugar. When the pan is hot, press the sugared side of the lemon onto the surface of the pan. Cook until the lemons start to caramelize, bubbling and sizzling, being careful not to burn.
Remove the lemon halves to a plate and cool completely, then squeeze ¼ cup of the juice into a small bowl. Save any leftover juice and lemons for later use.
Add a pinch of salt, Dijon mustard, and white wine vinegar to the bowl and whisk to combine. Slowly pour in the olive oil, whisking until the dressing is emulsified.