In a small bowl, combine the mayonnaise, minced garlic, lemon juice, salt, and black pepper. Stir until smooth. If you want an extra layer of flavor, add a teaspoon of cajun seasoning. Cover and refrigerate while you work on the chicken. This allows the flavors to meld.

Place one chicken breast on a cutting board. Hold your hand flat on top, and using a sharp knife, slice horizontally through the middle, almost but not all the way through, to open it like a book. Then, cut along the fold to create two even pieces. Repeat with the second breast. You will have four thin cutlets. This ensures even cooking and maximum crunch.

In a shallow bowl, whisk together the all-purpose flour, cornstarch, cajun seasoning, salt, and black pepper. In a separate bowl, whisk the egg and milk until combined. Have a large plate or baking sheet ready for the breaded chicken.

Take a chicken cutlet and dredge it first in the flour mixture, pressing gently to adhere. Shake off any excess. Then dip it into the egg mixture, letting the excess drip off. Finally, dredge it again in the flour mixture, pressing firmly so the coating sticks. Place on the prepared plate. Repeat with all cutlets. Let them rest for 5 minutes. This helps the breading set and creates a better crust.

In a large cast-iron skillet or heavy-bottomed pan, add about 1/2 inch of neutral oil (canola or vegetable). Heat over medium-high heat until it shimmers. To test, drop a pinch of flour into the oil; if it sizzles immediately, it is ready. The temperature should be around 350 degrees F.

Carefully place two breaded cutlets into the hot oil, making sure not to overcrowd the pan. Fry for 3 to 4 minutes per side, until deeply golden brown and the internal temperature reaches 165 degrees F. The kitchen will smell amazing, like southern fried chicken with a spicy twist. Transfer to a wire rack set over a baking sheet to drain. Sprinkle with a little extra salt while hot.

While the chicken rests, slice the brioche bun in half. Brush the cut sides with the melted butter. Place them in the same skillet (reduce heat to medium) and toast for about 1 minute, until golden. Set aside.

Spread a generous layer of the garlic aioli on the bottom half of each bun. Place a handful of lettuce leaves, then a slice or two of tomato. Lay the crispy chicken cutlet on top. If you like heat, add a few dashes of hot sauce. Top with the other half of the bun. Press gently and serve immediately.