Place each cutlet between two sheets of parchment and pound from the center outward until it's an even 4 mm thick, about 15 x 20 cm. Pat dry and season both sides with the salt and pepper.

Lay one slice of ham and one quarter of the fontina on one half of each cutlet, keeping a 1 cm border clear. Fold the empty half over, press the edges firmly, and pin each cutlet closed with 2 toothpicks.

Set the flour, beaten eggs, and breadcrumbs in three shallow dishes. Dredge each cutlet in flour and shake off the excess, dip it in egg, then press it into the breadcrumbs on both sides and along the edges until fully coated.

Place the cutlets on a parchment-lined plate and chill uncovered for 10 minutes. Meanwhile, heat the oven to 100 C / 212 F and set the wire rack in the sheet pan.

Heat the oil and butter in the 30 cm / 12-inch skillet over medium heat for about 3 minutes, until a breadcrumb sizzles on contact (about 170 C / 340 F). Fry 2 cutlets for 3 to 4 minutes per side, until the crust is deep golden. Move them to the rack in the oven, where they can wait up to 15 minutes.

Skim loose crumbs from the oil and fry the remaining 2 cutlets the same way. Slide a thermometer in from the edge of one cutlet, which should read 145 F / 63 C. Remove all toothpicks and serve at once with lemon wedges.