Using a mallet, pound the chicken to an even thickness.
In a large bowl, resealable bag (or a shallow bowl), whisk together the balsamic vinegar, lemon juice, olive oil, garlic, honey (if using), rosemary, basil, salt and pepper. Add the chicken and toss until coated with the marinade. Gently squeeze the air out of the bag. Marinate for 30 minutes or up to 4 hours.
Preheat a grill to medium-high heat (375 - 450 degrees Fahrenheit). Carefully grease the cooking grates with the oil. If using a grill pan, heat the pan over medium-high heat and add a little oil to the grill pan.
Remove the chicken from the marinade. Discard the marinade. Grill the chicken for 3 to 4 minutes per side or until cooked through. Cooking times may vary depending on the thickness of the chicken and your grill's heat.
Remove the chicken from the grill. Allow chicken to rest for 5 to 10 minutes before slicing or serving.
Notes
Pound the chicken evenly: Chicken breasts cook more evenly when they are similar in thickness, preventing dry edges and undercooked centers.
Don't marinate too long: Because the marinade contains both balsamic vinegar and lemon juice, avoid marinating longer than 4 hours or the texture of the chicken can become mushy. Do not freeze raw chicken in the marinade. While defrosting the meat will become mushy due to the high acid in the marinade.
Let the chicken sit at room temperature briefly: If marinated in the refrigerator, allow the chicken to sit at room temperature for about 20 minutes before grilling for more even cooking.
Make Ahead: The marinade can be prepared up to 2 days ahead and stored refrigerated until ready to use.
Storing: Store leftover grilled chicken in an airtight container in the refrigerator for up to 3 days.
Freezing: Cooked grilled chicken can be frozen for up to 3 months.