Place one chicken breast inside a food storage bag or under plastic wrap and pound the thicker part of the breast with the flat side of a meat mallet or large skillet to flatten. The chicken will cook evenly when it is an even thickness.
Whisk together the marinade ingredients in a medium bowl and transfer to a large resealable plastic bag.
Add the chicken, making sure the marinade surrounds each piece, seal the bag and refrigerate for 1 to 4 hours. Set the butter for the compound butter out on the counter to soften to room temp.
When ready to grill, start with the tomatoes for the compound butter. Remove chicken from marinade and transfer to a platter while preparing the butter.
Heat the grill to 400°F.
For the Compound Butter:
Toss the tomatoes with 1 tablespoon of the oil to coat and place on skewers, or use a vegetable grill pan. If using a grill pan, place on the grill while preheating so it is hot when the tomatoes go on it.
Grill the tomatoes until blistered and charred in places, about 8 minutes. Remove from grill and transfer to a cutting board. Sprinkle with salt and pepper and coarsely chop. Transfer to a mixing bowl and place in the refrigerator for 5-10 minutes to cool.
Combine with the remaining 2 tablespoons of olive oil, basil, and softened butter until well mixed. Spoon compound butter onto a piece of wax paper in a long, narrow shape, about 2 inches wide by 9 inches long.
Carefully fold the wax paper over the butter to create a somewhat rounded cylinder shape. Gently place the butter on a flat surface in the freezer while grilling the chicken. (The butter will harden enough to slice and top the chicken.)
To Grill the Chicken:
Turn one side or one burner of the grill off to create an indirect heat zone (adjust the other burners to maintain the temperature around 400°F). Place the chicken on the indirect side and close the lid.
Allow to cook for 4-6 minutes on each side, or until the internal temperature of the chicken is at least 130°F. Move the chicken to directly over the heat to create grill marks and finish cooking, about 2 more minutes per side.
Remove from grill when the internal temperature of the thickest part is between 153-155°F and tent loosely with foil to rest for 5 minutes.
Remove the tomato-basil butter from the freezer (or from the fridge if it was made in advance). Slice into approximate 1 tablespoon-sized pieces and place one on top of each piece of chicken while the chicken is tented and resting;
butter should melt over the chicken. If the butter is still not melted when the chicken is done resting, place in a warm oven for a couple of minutes until ready to serve.
Prep time includes marinating time.
Tomato Basil Compound Butter can be made well in advance and refrigerated until ready to use, for up to 2 weeks. This recipe will only require about half of the butter made, the remaining butter can be stored in the freezer for other uses such as topping grilled or roasted meats, spread for breads (allow to soften), or a starter for pan sauces.
Change up the flavor with 3 tablespoons Savoring Today's Green Chile Chix Rub in the marinade and 2-3 tablespoons of chopped green chiles with 1 tablespoon cilantro mixed into the compound butter.