Make the marinade. Cut all of the peaches in half, remove the pits, and add them to a powerful blender or food processor. Add the basil leaves, water, sugar and garlic and purée on high until you have a relatively smooth consistency.
Prepare the chicken. Trim any excess fat off of the chicken breasts and then put them in a large, heavy-duty zip-lock bag. Set aside.
Marinate. Pour about half of the peach mixture into the bag with the chicken breasts. Move the chicken around to be sure all sides are well coated. Remove the air from the bag, seal, and refrigerate for at least 6 hours and ideally overnight. (Save the other half of the peach mixture in a tightly sealed container in the refrigerator for the sauce.)
Grill. When you're ready to cook, preheat your stove-top grill or outdoor BBQ. Remove the now marinated chicken breasts from the bag and place them on a large plate. (Discard the bag with any excess marinade.) Once the grill is hot, season both sides of each chicken breast with salt and pepper, and then place them on the grill. (You should hear a sizzling sound when they hit the grill -- if you don't, it's not hot enough. Wait for the sizzle!) Grill just until the chicken is cooked through, about 4 minutes per side.
Make the sauce. While the chicken is grilling, add the remaining peach mixture (from the container in the fridge), to a small sauce pan. Add the lemon juice, bring to a simmer, and let it thicken a bit. This should take about 5 minutes. Season to taste with a bit more brown sugar, salt and pepper. (Here's How to Season to Taste.) Serve. Serve the chicken either whole or sliced, placing it on top of the sauce, on a platter or serving plates. Garnish with fresh basil leaves and peach slices if desired.