Make the pimento cheese: Combine the cheddar, jack, and cream cheese in a large bowl. Add the diced pimentos, mayonnaise, mustard, 2 teaspoons adobo sauce, and ¼ teaspoon pepper and stir to combine. Set aside.
Fry the chicken: Heat 1½ inches vegetable oil in a large heavy-bottomed pot over medium-high heat until a deep-fry thermometer reaches 350˚F.
Whisk together the flour, cornstarch, salt, garlic powder, onion powder, and remaining ¾ teaspoon pepper in a shallow dish. Whisk the buttermilk and remaining 2 tablespoons adobo sauce in a separate shallow dish.
Cut the chicken breasts in half horizontally to make 4 thin cutlets. Working with one piece at a time, dredge the chicken in the flour mixture, shaking off any excess. Dip in the buttermilk, letting the excess drip off. Dredge again in the flour mixture, pressing well to thoroughly coat.
Working in 2 batches, fry the chicken until golden brown and cooked through, about 3 minutes per side. Remove to a rack and sprinkle with salt.
Spread some pimento cheese on the cut sides of each bun. Fill with the chicken, pickle slices, and lettuce. Serve with potato chips.