1Marinate the chicken — Whisk 1/3 cup orange juice, 3 tbsp lime juice, 4 garlic cloves, 1 tsp cumin, the oregano, salt, and 2 tbsp olive oil. Add the chicken and marinate for 20 minutes at room temperature, or up to 2 hours in the refrigerator. Set the remaining orange and lime juice aside for the sauce.
2Sear the thighs — Remove the chicken from the marinade, discard the marinade, and pat the thighs very dry. Heat a cast-iron skillet over medium-high heat with a thin film of oil. Sear the smooth side down, undisturbed, for 5-6 minutes. Flip and cook for 4-5 minutes more, until the internal temperature reaches 175°F.
3Rest and slice — Rest the chicken for 5 minutes, then slice it 1/4 inch thick across the grain.
4Make the mojo butter — Reduce the heat to medium-low. Cook the remaining 2 garlic cloves in the chicken fat for 30 seconds. Add the reserved citrus juice, stock, and remaining 1/2 tsp cumin, scraping up the fond. Reduce by about half, 3-4 minutes. Remove the pan from the heat and swirl in the cold butter, one cube at a time, until glossy. Pour the sauce into ramekins and keep warm; do not let it boil again. Wipe out the pan.
5Build the sandwiches — Split the rolls and spread mustard on both cut sides. Layer each sandwich with Swiss cheese, chicken, pickles, Swiss cheese, and the top bun. Butter the outsides of the rolls.
6Press the sandwiches — Set the cast-iron skillet over medium heat. Press two sandwiches at a time under a second heavy skillet or foil-wrapped brick for 3-4 minutes per side, until deep golden and the cheese melts. Lower the heat if the bread browns before the cheese melts.
7Plate and serve — Spread aioli across each plate. Cut the sandwiches on a hard diagonal and stack the halves with the cross-sections facing out. Serve with fries and a ramekin of mojo butter for dipping.
Original build, adapted from a traditional Cubano with the ham omitted on purpose. If it needs more punch, add more pickles rather than a substitute protein.
Never reuse the marinade in the sauce, because it has held raw chicken. The sauce uses the reserved fresh juice.
Rewarm the mojo butter gently or it will split. If it does split, whisk in a splash of warm stock.
Leftover pressed sandwiches go soggy. For better leftovers, store the sliced chicken separately and press fresh.