To make the dipping sauce, in a bowl, stir together the apricot preserves, soy sauce, ketchup, and Worcestershire sauce. Divide amont 4 small bowls or ramekins (optional- I just put it in one bowl and share with my husband).
Season the chicken breasts with salt and pepper. Spread the flour on a plate, beat the eggs in a shallow bowl until blended, and spread the bread crumbs on another plate. One at a time, dredge each chicken breast in the flour, dip in the eggs, and then coat evenly with the crumbs. Set aside on a rack or baking sheet.
In a large frying pan, heat the oil over medium-high heat until very hot but not smoking. Add the chicken breasts and reduce the heat to medium. Cook, turning once, until the crumbs are golden brown and the chicken breasts feel firm when pressed in the centers, about 4 minutes on each side. Adjust the heat as needed so the crumbs don't burn. Transfer to a cutting board.
Cut each chicken breast across the grain into wide strips. Slide a knife under the strips and transfer to warmed individual plates. Serve immediately with the dipping sauce.