The night before, make the brine: stir 1/4 cup kosher salt and 2 tablespoons sugar into 2 cups water until dissolved.
Place the chicken breasts in a zipper bag or covered container, pour the brine over them so they are fully submerged, seal, and refrigerate for 12 hours.
The next day, remove the chicken from the brine, rinse under cold water, and pat dry with paper towels.
Between two sheets of plastic wrap, gently pound each breast to an even thickness of about 1/2 inch; remove the plastic wrap when done.
Sprinkle 2 tablespoons McCormick Cajun Seasoning evenly over both sides of the chicken and rub it in with your fingers.
Heat a large skillet over medium for about 2 minutes. Spread 2 teaspoons softened butter on the cut sides of the brioche buns and toast butter-side down in the skillet until golden; transfer to a plate.
In the same skillet, heat 4 tablespoons olive oil about 30 seconds. Add the seasoned chicken breasts and cook undisturbed 5 to 6 minutes on the first side until a dark crust forms, then flip and cook 5 to 6 minutes more until an instant-read thermometer reads 165°F.
Transfer the cooked chicken to a paper-towel-lined plate and let rest 2 to 3 minutes.
Spread 3 tablespoons mayonnaise on the cut sides of each toasted bun, top the bottom bun with sliced dill pickles, place a chicken breast on top, and close with the top bun; serve immediately.