In a mixing bowl, dissolve the 2 chicken bouillon cubes and 1/4 teaspoon seasoned salt into 2 cups cool water; add the chicken breasts, cover, and refrigerate for 12 hours or overnight to brine.
After brining, pour off and discard the marinade and pat the chicken breasts lightly to remove excess liquid.
In a shallow dish, combine 1 cup all-purpose flour, 1 1/2 cups finely crushed saltine crumbs, 2 teaspoons powdered sugar, and 1/4 teaspoon paprika; stir to blend evenly.
Dredge each chicken breast in the seasoned flour mixture, pressing the coating onto the surface, then place the breaded breasts on a wire rack and let them rest a few minutes to set.
Heat peanut oil (or vegetable oil) in a deep-fryer, large pot, or deep skillet to 350°F (175°C); use enough oil for about 4 inches depth if using a pot.
Fry the chicken breasts 7 to 8 minutes per side as needed until golden brown and an instant-read thermometer inserted into the thickest part reads 165°F (74°C); adjust time for thickness and drain on a clean wire rack.
Melt the butter and brush it on the inside of each hamburger bun.
Assemble each sandwich by placing one fried chicken breast on the bottom bun, topping with two dill pickle slices, and finishing with the top bun.