Make the dipping sauce: in a bowl combine the apricot preserves, natural ketchup, soy sauce and Worcestershire sauce; stir until smooth. Divide the sauce among four small dipping bowls and set aside.
Set up three dredging stations: spread the 1/3 cup all-purpose flour on a plate, beat the 2 large eggs in a shallow bowl, and spread the 1 1/2 cups panko bread crumbs on another plate.
Dredge each chicken breast in this order: coat lightly with flour and shake off any excess, dip into the beaten eggs, then coat evenly with the panko bread crumbs. Take care not to press the breadcrumbs into the chicken.
Pour 1/4 cup oil into a large frying pan and heat over medium-high until the oil is very hot but not smoking.
Add the breaded chicken breasts to the pan, reduce the heat to medium, and fry about 4 minutes on the first side. Flip and fry about 4 minutes on the second side, or until the crust is golden brown and the chicken feels firm when pressed in the center. Reduce the heat if the crumbs start to burn or brown too quickly.
Transfer the cooked chicken breasts to a cutting board and slice each breast across the grain into wide strips.
Transfer the sliced chicken to individual serving plates and serve immediately with the apricot dipping sauce.