Place a large, heavy-bottomed saucepan or Dutch oven over medium-high heat, carefully add the oil, and heat until it registers 350°F (180°C) on an instant-read deep-fry or candy thermometer.
Place a rimmed baking sheet alongside the stove and on it place a wire cooling rack, a brown paper grocery bag, or several thicknesses of paper towels. Preheat the oven to 200°F (90°C).
While the oil comes to temperature, in a shallow bowl, combine the flour and black pepper. When the oil is hot, remove one chicken thigh at a time from the marinade, allowing any excess to drip off, and dip it in the flour mixture, turning to completely coat it. Shake off any excess flour. Then repeat, dipping each piece back in the marinade and then in the flour mixture again. Place the chicken on a plate when done.
Place a couple pieces of chicken in the hot oil, taking care not to overcrowd the pan. The chicken should be fully covered by the hot oil. Fry the chicken, turning occasionally, until the chicken is golden brown and the thermometer inserted in the thickest portion of the thigh registers 165°F (75°C), 5 to 10 minutes. Allow the oil to return to 350°F before cooking another batch of chicken. (If you don’t have a thermometer handy, use the tip of a sharp knife to cut into the thickest portion of a chicken thigh to check for doneness—there should be no trace of pink.)
Using tongs, transfer the fried chicken to the rack or brown paper or paper towels on the baking sheet. If not devouring the chicken straight out of the oil, transfer the baking sheet to the preheated oven to keep warm.