Melt butter in a small saucepan over medium heat. Add shallot and cook until softened, about 3 minutes. Stir in brown sugar, bourbon, vinegar, mustard, 1/2 teaspoon kosher salt and chopped peaches.
Bring to a gentle simmer and cook, stirring occasionally, until peaches are soft and the sauce has thickened, 15-20 minutes. Keep warm.
Turn all burners on gas grill to high, cover, and heat until hot, about 15 minutes. Leave primary burner on high and turn other burners to medium.
Combine brown sugar, ginger, salt and pepper in a small bowl. Pat pork chops dry and coat all over with sugar mixture. Oil grates and place chops on hot side of grill. Cook 8-10 minutes, flipping halfway, until browned on both sides.
Move to cooler side of grill and brush with melted butter. Continue to cook, turning once and brushing with more butter, until an instant-read thermometer registers 145°F when inserted into the center of the chop, about 5 more minutes. Transfer to a platter and spoon warm bourbon peach sauce over the meat.