1Halve chicken breasts through to make 8 cutlets; pound very thin, about 1/4 inch. Season the chicken on both sides with salt and pepper. Spread a thin layer of Dijon on one side of the cutlets and sprinkle with thyme. Fold and arrange 1 slice each of cheese and ham on 4 cutlets, leaving a little chicken exposed all around. Place the remaining cutlets Dijon-side down over the top and press down at the edges to form 4 stuffed cutlets. Dredge in flour.
Beat eggs and lemon juice together in a shallow dish.
Heat a large skillet with oil, 2 turns of the pan, over medium to medium-high heat. Melt 1 tablespoon butter into oil.
2Turn 2 stuffed cutlets in egg batter and add to pan. Cook about 8 minutes, turning occasionally or until cooked through and golden. Transfer to a platter and cover loosely with foil. Melt another tablespoon into the pan and brown remaining cutlets. Remove to the platter.
3Add remaining butter to pan and sprinkle in flour 1 minute, whisk in stock and thicken to form gravy; season with salt and pepper. Whisk in maple syrup or honey and mustard.
Halve stuffed breasts, top with gravy and serve.