Slice chicken breasts in half lengthwise to make 4 thinner pieces. Pound to even thickness.
In a bowl, whisk Dijon mustard, balsamic vinegar, juice from the lemon, garlic, Italian seasoning, olive oil, and salt and pepper to taste. Set aside 1/3 cup marinade for later use.
Add the remaining marinade and chicken to a resealable plastic bag. Seal and refrigerate for 30 minutes to 3 hours.
Clean and oil grill grates. Preheat grill to medium heat (400–450°F).
Grill chicken 4–6 minutes per side or until internal temperature reaches 160°F. Brush chicken with 2 tablespoons of the reserved marinade while grilling.
Drizzle remaining marinade over chicken before serving.