Heat a skillet with butter over medium heat. Add peaches and the juice of 1/2 lemon, cook until tender and lightly golden, 10 minutes. Turn off heat.
While peaches cook, heat 1 tablespoon EVOO, a turn of the pan, in a large skillet over medium high to high heat. Cut chicken breast pieces in half across the center of each breast to make 4 equal portions. Add chicken breasts and thighs to pan, season with salt and pepper. Cook 12 minutes, remove to plat and cover with foil.
Drizzle the pan with the remaining extra-virgin olive oil, half a turn of the pan. Add the chopped shallot, and the ginger, and cook for 1 to 2 minutes. Stir in the stock, the preserves, hot sauce and Worcestershire sauce and season with black pepper, to taste. Cook over medium heat for a couple of minutes to thicken. Arrange the chicken and peaches on a platter and douse with peach sauce. Portions are 1 piece of light meat and 1 thigh per person.