In a large bowl, whisk together the buttermilk, shallot, garlic, jalapeño, 2 teaspoons kosher salt, and 1 teaspoon black pepper. Place the chicken breasts between 2 pieces of parchment paper and, with a rolling pin or a meat mallet, pound them until they are evenly ½ inch thick. Slice each piece of chicken in half crosswise so you have 2 pieces approximately the same size. Place the chicken in the marinade, making sure each piece is well coated, cover with plastic wrap, and refrigerate for at least 8 hours or up to (but not more than) 24 hours.
When ready to cook the chicken, preheat the oven to 300 degrees. Set a wire rack on a sheet pan and place them in the oven.
In a large bowl, whisk together the flour, baking powder, paprika, cayenne pepper, celery salt, 1 tablespoon kosher salt, and 1½ teaspoons black pepper and set aside.
Pour oil into a medium (9-inch round × 4½-inch high) Dutch oven, such as Le Creuset, until it is 2 inches deep. Clip on a candy thermometer and heat the oil over high heat until it reaches 350 degrees. Meanwhile, lift the chicken from the marinade, dredge it in the flour mixture, submerge it again in the marinade, then again in the flour mixture, lightly shaking off the excess. Transfer the chicken to a plate or sheet pan until ready to fry.
When the oil is 350 degrees, carefully lower 3 pieces only into the oil with tongs and adjust the heat to keep the oil at 350 degrees. Don’t crowd the chicken! Cook for 5 minutes, turning once to brown evenly. With a slotted spoon, transfer the chicken to the sheet pan in the oven to keep warm. Repeat with the remaining chicken. Sprinkle all the chicken with salt and keep warm for up to 15 minutes, until ready to serve.
To assemble, place the bottom of each bun on a plate, spread with some of the Buttermilk Herb Mayo, then a lettuce leaf, 4 pickle slices, then a piece of chicken. Spread the underside of the top bun generously with more of the mayo and place on top of the chicken, mayo side down. Serve while the chicken is still warm.
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