Prepare the chicken: Place chicken pieces between plastic wrap and pound them to an even 1/2-inch thickness.
Mix the dry coating: In a shallow dish, whisk together the flour, cornstarch, paprika, cayenne, salt, and pepper.
Mix the wet dip: In a second shallow dish, whisk the eggs and buttermilk together.
Coat the chicken: Dip each piece of chicken first into the wet mixture, letting excess drip off, then press firmly into the dry mixture to coat completely. Repeat the wet-then-dry dip for a double coating. Set aside on a rack.
Make the aioli: Squeeze the soft, roasted garlic pulp from the head into a small bowl. Mix the garlic pulp, mayonnaise, lemon juice, and salt until smooth.
Heat the oil: Pour vegetable oil into a deep, heavy-bottomed pot or Dutch oven to a depth of about 1.5 inches. Heat the oil to 350°F (175°C).
Fry the chicken: Carefully place 1 or 2 chicken pieces into the hot oil, ensuring you do not overcrowd the pot. Fry for 4 to 6 minutes per side, until golden brown and the internal temperature reaches 165°F (74°C).
Drain: Remove the chicken and place it on a wire rack lined with paper towels to drain excess oil. Season immediately with a pinch of salt.
Toast the buns: Lightly butter the cut sides of the brioche buns and toast them in a dry skillet over medium heat until golden brown.
Assemble the sandwich: Spread a generous amount of roasted garlic aioli on both halves of the toasted bun. Place lettuce, the crispy chicken fillet, cheese (if using), and pickles on the bottom bun. Top with the other bun half.
Notes
To roast the garlic, slice the top off the head, drizzle with a little oil, wrap in foil, and bake at 400°F (200°C) for 30 to 40 minutes until soft.
For extra crispiness, let the coated chicken rest in the refrigerator for 15 minutes before frying.
If you prefer an air fryer method, spray the coated chicken with cooking spray and cook at 380°F (195°C) for 15-18 minutes, flipping halfway.