In the morning, combine 2 tablespoons salt and a cup of warm water in a large bowl or container. Stir to dissolve the salt. Trim chicken of excess skin and fat. Add chicken to the bowl. Cover with very cold water and add a tray of ice cubes. Mix everything together with your hands. Cover and chill in the refrigerator until dinner time.
Preheat oven to 400°F. Remove chicken from the fridge and pat pieces dry completely with paper towels. Put the butter in a rimmed roasting pan large enough to fit the chicken in one layer without crowding it—we used a hotel sheet pan. Place the pan in the oven.
In a 1-gallon plastic bag, combine the flour, remaining 1 tablespoon salt, 1 teaspoon of pepper, lemon zest and Parmesan. Give it a good shake. Add the chicken pieces two at a time and shake them until thoroughly coated. Remove them from the bag, shaking vigorously to remove excess flour. Transfer to a plate, and flour the rest.
Lay the chicken thighs in the roasting pan, skin side down, and oven-fry until brown and crisp on the bottom, about 40 minutes (it may take longer). When nice and brown, use a spatula to scrape them up off the pan and turn them. Cook the other side until browned, which will take less time, around 10 to 20 minutes. Transfer to a plate lined with paper towels. Before serving, season with a little more fresh pepper and Kosher salt.