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For the marinade, in a medium bowl, whisk together the buttermilk, shallot, garlic, jalapeño, 2 teaspoons salt, and 1 teaspoon black pepper.
With a rolling pin or a meat mallet, pound the chicken breasts between 2 sheets of parchment paper until they are evenly 1/2 inch thick. Slice each chicken breast in half crosswise so you have 2 pieces approximately the same size.
Add the chicken to the marinade, cover with plastic wrap, and refrigerate for at least 8 hours but not more than 24 hours.
Meanwhile, in a medium bowl, whisk together the flour, baking powder, paprika, cayenne pepper, celery salt, 1 tablespoon salt, and 1 1/2 teaspoons black pepper and set aside.
When ready to cook the chicken, preheat the oven to 250 degrees. Place a sheet pan with a baking rack on it in the oven.
Pour the oil into a medium (9-inch round x 4 1/2-inch high) Dutch oven, such as Le Creuset, fitted with a candy thermometer, until the oil is 2 inches deep. Heat the oil over high heat until the temperature reaches 350 degrees.
Meanwhile, lift the chicken from the marinade, dredge it well in the flour mixture, submerge it again in the marinade, then dredge again in the flour mixture, dusting off the excess. Transfer the chicken to a plate or sheet pan until ready to fry.
When the oil is 350 degrees, lower three pieces only(!) carefully into the oil with a slotted spoon and adjust the heat to keep the oil at 350 degrees. Don't crowd the chicken! Cook for 5 minutes exactly, turning once to brown evenly. With the slotted spoon, transfer the chicken to the sheet pan in the oven to keep warm. Fry the remaining three pieces, sprinkle all the chicken with salt, and serve hot.