Marinate — the night before. Whisk the buttermilk, shallot, garlic, jalapeño, salt, pepper, and hot sauce in a large bowl. Pound each chicken breast between parchment to an even 1/2-inch thickness, then cut crosswise into two roughly equal pieces — six total. Submerge, cover, and refrigerate 8 to 24 hours. Don't exceed 24 or the texture turns mealy.

Make the herb mayo. Whisk everything together, season, and refrigerate at least an hour. Better overnight, and it keeps about five days.

Set up your station. Whisk the dredge ingredients in a wide shallow dish. Set a wire rack over a sheet pan. Pour 2 inches of oil into a Dutch oven and heat to 350°F. Heat the oven to 200°F with a second rack-lined sheet pan inside.

Dredge, twice. Lift a piece from the marinade and let the excess drip off. Press firmly into the flour on both sides, dip back into the marinade, then back into the flour, pressing hard so the coating grips. Rest the coated pieces 10 minutes so the coating hydrates and won't slide off in the oil.

Fry in batches. Working two at a time, lower the chicken into the oil. Fry 3 to 4 minutes per side until deep golden and 160°F at the center. Don't crowd the pot — let the oil return above 325°F between batches. Hold on the rack in the warm oven up to 15 minutes.

Toast and assemble. Butter the cut sides of the buns and toast in a dry skillet until golden — a moisture barrier, not a garnish. Spread herb mayo on both halves. Bottom bun, pickles, chicken, lettuce, top bun. Serve immediately.