Heat a medium skillet with the butter over medium heat. Add the peaches and the juice of 1/2 lemon, and cook until tender and lightly golden, 10 minutes. Turn off the heat.
While the peaches cook, heat 1 tablespoon extra-virgin olive oil, a turn of the pan, in a large skillet over medium-high to high heat. Cut the chicken breasts in half across the center of each breast to make 4 equal portions. Add the chicken breasts and thighs to the pan and season with salt and pepper, to taste. Cook until the chicken is cooked through, about 12 minutes, then remove to a plate 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.