In a large bowl, stir together 10 cups ice water and 2 tablespoons coarse salt. Add chicken, submerging all pieces. Cover; let soak overnight in refrigerator.
In another bowl, whisk together buttermilk, 1 tablespoon coarse salt, mustard, and 1/2 teaspoon each cayenne and black pepper. Transfer chicken to marinade, submerging all pieces. Cover; refrigerate at least 4 hours and up to overnight.
Let chicken stand at room temperature 1 hour. In a shallow dish, whisk together flour, cornmeal, remaining 1 teaspoon each coarse salt and black pepper, and remaining 1/2 teaspoon cayenne.
Heat a heavy pot filled halfway with oil over medium heat until a deep-fry thermometer reads 350 degrees. Remove chicken from marinade, a few pieces at a time (do not shake off excess); dredge in flour mixture. Transfer to a baking sheet.
Fry chicken in batches until golden brown and a thermometer inserted into thickest part (avoiding bone) reads 160 degrees, 6 to 7 minutes for small pieces, 9 to 10 minutes for large. Adjust heat to maintain a temperature between 340 degrees and 350 degrees during cooking. Transfer to a wire rack set on a rimmed baking sheet; season with sea salt. Serve warm or at room temperature.