Make the dipping sauce: in a medium bowl combine 3/4 cup apricot preserves/jam, 3 tablespoons natural ketchup, 3 tablespoons soy sauce, and 1 teaspoon Worcestershire sauce. Stir until smooth, then divide the sauce among four small dipping bowls and set aside.
Prepare the dredging stations: spread 1/3 cup all-purpose flour on a plate, crack and beat 2 large eggs in a shallow bowl, and spread 1 1/2 cups panko bread crumbs on a separate plate. Confirm the 4 small boneless skinless chicken breasts are pounded to 1/4-inch thickness.
Bread the chicken, one breast at a time: dredge a chicken breast in the flour and shake off any excess, dip it into the beaten eggs until coated, then transfer it to the panko and coat evenly. Do not press the panko into the chicken; just turn the breast to coat all sides. Repeat with the remaining breasts.
Heat the oil: pour 1/4 cup oil for frying into a large frying pan and heat over medium-high until the oil is very hot but not smoking.
Fry the chicken: add the breaded chicken breasts to the pan (do not crowd; cook in batches if needed) and immediately reduce the heat to medium. Fry about 4 minutes on the first side, then flip and fry about 4 minutes on the second side, until the crust is golden brown and the chicken feels firm when pressed in the center. Reduce the heat if the crumbs are browning too quickly.
Rest and slice: transfer the cooked breasts to a cutting board and let rest briefly. Slice each breast across the grain into wide strips.
Serve: arrange the chicken strips on individual plates and serve immediately with the apricot dipping sauce.