Whisk the wet ingredients. In a large bowl, whisk together the egg, mayonnaise, Dijon, Worcestershire, lemon juice, Old Bay, parsley, salt, and pepper until smooth.
Fold in the crab and panko. Add the drained lump crab and panko. Gently fold with a rubber spatula — do not stir or smash. You want big visible chunks of crab. If the mixture feels too wet to hold a patty, add another tablespoon of panko.
Shape and chill. Divide the mixture into 4 equal patties, about ¾ inch thick. Place on a parchment-lined plate and refrigerate for at least 15 minutes. (Don't skip — this keeps them from falling apart.)
Preheat the air fryer to 400°F for 3 minutes.
Air fry. Lightly spray the air fryer basket. Place the crab cakes in a single layer with space between them and spray the tops with oil. Air fry at 400°F for 5 minutes. Carefully flip with a thin metal spatula, then air fry another 4–5 minutes, until the crust is deep golden brown and the internal temperature reaches 145°F.
Toast the buns. In the last 2 minutes of cooking (or after removing the crab cakes), place the split brioche buns cut-side up in the air fryer for 1–2 minutes until lightly toasted.
Assemble. Spread remoulade on the bottom bun, layer with butter lettuce, the crab cake, a tomato slice, more remoulade on the top bun, and close the sandwich. Serve immediately with lemon wedges.