Halve 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.