Place all of the remoulade ingredients in a bowl and whisk well to combine. Taste and adjust for seasonings. Set aside in the refrigerator.
Stir together all of the crab cake ingredients except for the crabmeat, breadcrumbs, and oil in a large mixing bowl until well combined.
Gently fold in the crabmeat and breadcrumbs until evenly coated. Be careful not to break up any lumps.
Form the mixture into six patties.
Grease a baking sheet with oil.
Move an oven rack to a position 6 to 8 inches from the heat source and preheat the broiler on high.
Place the crab cakes on the baking sheet and broil for 4 to 5 minutes per side, until browned and heated through. Watch them carefully so they don't burn.
Remove the crab cakes from the oven and place them on buns with lettuce and a tomato slice underneath.