Bring a large pot of salted water to a boil. Peel and quarter the potatoes, then cook until fork-tender, about 20 to 25 minutes. Drain, reserving 1/2 cup of the cooking water.
Pass the potatoes through a ricer or mash them through a fine mesh sieve into a mixing bowl. Stir in the olive oil and salt. Add a little reserved cooking water, a tablespoon at a time, until smooth and creamy. Set aside to cool.
Trim and clean the chicken breasts. Butterfly each piece by slicing horizontally almost all the way through, then open like a book. Make an additional cut from the center outward to expand the surface. Cover with plastic wrap and pound to an even 1/4-inch thickness using a meat mallet or rolling pin.
Place 2 heaping tablespoons of mashed potato on one half of each chicken cutlet, spreading it evenly and leaving a 1/2-inch border. Top with a second cutlet, pressing the edges to seal.
Prepare three shallow bowls. In the first, place the flour. In the second, whisk the eggs with mayonnaise, garlic, and mustard. In the third, combine the golden breadcrumbs, panko, paprika, cumin, salt, and sesame seeds.
Dredge each stuffed schnitzel in flour, then dip into the egg mixture, followed by the breadcrumb mixture, pressing gently to coat thoroughly.
Heat 1/2 inch of neutral oil in a large skillet over medium heat. Fry each schnitzel until deeply golden and crisp, about 4 to 5 minutes per side. Transfer to a wire rack to drain.
Season immediately with flaky salt and serve hot with lemon wedges and a fresh salad.