Place chicken thighs between parchment and press until evenly flattened, about ¼-inch thick.
Mix buttermilk, hot sauce, and salt. Submerge chicken and marinate at least 10 minutes (overnight if you have time).
Mix flour, potato starch, paprika, pepper, and salt. Press chicken firmly into the dredge, squeezing and roughing up the edges.
Heat neutral oil to 350°F / 175°C. Fry chicken until deeply golden, 3–4 minutes per piece. Drain on a rack.
Combine Nashville hot spices in a heatproof bowl. Whisk in hot fryer oil until smooth, then toss chicken to coat.
Whisk sour cream, mayo, milk, dill, pickles, salt, and pepper until smooth.
Butter a skillet over medium heat. Add sourdough and pile on white cheddar, letting some spill to form a cheese crust. Add chicken, then transfer to a 400°F / 205°C oven for 3–4 minutes until melted and crisp.
Spoon remoulade over the chicken, close the sandwich, press lightly, slice, and serve.