In a medium mixing bowl, mash the anchovy filets with the oil they're packed in. Stir in the cream cheese, lemon juice, Worcestershire sauce, vinegar, ground mustard, Dijon mustard, and Parmesan cheese until well combined. Gently fold in spinach.
Make a pocket in each chicken breast by inserting a paring knife into the side of the thickest part and working the knife around inside the meat to create a cavity. Generously stuff the cream cheese mixture into the cavities, then secure the openings with toothpicks.
Put the beaten eggs in a shallow bowl or plate. Put the panko crumbs in a second shallow bowl or plate.
Heat the oil in a large skillet over medium-high heat. Carefully dip the stuffed chicken breasts in the beaten egg, then coat with the panko crumbs. Cook in the hot oil until browned on both sides, approximately 5 minutes per side. Transfer to an oiled 9 x 13 inch baking pan.
Bake uncovered in the preheated oven until cooked through, about 20 minutes. Remove from oven and let stand 5 minutes before serving.