Marinate the Chicken: Begin by preparing the marinade for the chicken. In a small bowl, combine balsamic vinegar, freshly grated ginger, and a pinch of salt.
Season the Chicken: Generously season the chicken with salt and place it in a large bowl or resealable bag. Pour in 2 tablespoons of the balsamic mixture, ensuring that the ginger is evenly distributed by stirring it before measuring. Retain the remaining balsamic mixture for later use. Add thyme, garlic, soy sauce, and lime juice to the chicken, then cover or seal. Allow the chicken to marinate in the refrigerator for a minimum of 2 hours, or up to overnight for enhanced flavor.
Prepare the Grill: Preheat the grill or broiler, positioning the rack approximately 4 inches from the heat source.
Grill the Peaches: In a separate bowl, mix together melted butter, thyme (or rosemary), and honey. Brush the peaches lightly with the butter mixture, then place them on a grilling basket or directly on the grill. Grill the peaches for 2 to 4 minutes per side, until the skin begins to curl back and the flesh becomes tender and jammy. Transfer the peaches to a serving platter and, if desired, drizzle them with additional butter mixture and sprinkle with flaky sea salt.
Grill the Chicken: If space permits, grill the chicken alongside the peaches or wait until the peaches are done. Remove any excess marinade from the chicken, pat it dry, and lightly coat it with olive oil. Grill or broil the chicken for 4 to 6 minutes per side until charred and browned.
Serve: Once cooked, transfer the chicken to a serving platter alongside the grilled peaches. Optionally, serve with a dollop of yogurt on the side, a drizzle of the remaining balsamic glaze, and a splash of olive oil. Garnish with scallions and basil if desired.