Start by making the crab cake mixture. In a large bowl, whisk together the mayonnaise, egg, Dijon mustard, lemon juice, Old Bay seasoning, salt, and pepper until smooth. This is your flavor base.
Gently fold in the crab meat, being careful not to break it up too much. Big chunks of crab are what make these cakes special. Add the breadcrumbs and parsley, folding just until everything comes together. The mixture should be moist but able to hold its shape.
Cover the bowl and let the mixture rest in the refrigerator for about 20–30 minutes. This step helps the crab cakes firm up and makes them easier to cook.
While the mixture chills, prepare the sauce by mixing all the sauce ingredients in a small bowl. Taste and adjust seasoning if needed. Set aside in the fridge.
Once chilled, form the crab mixture into four large patties. Heat butter or oil in a skillet over medium heat. Carefully place the crab cakes in the pan and cook for about 4–5 minutes per side, until golden brown and heated through.
Remove the crab cakes and let them rest briefly on paper towels. Toast the buns lightly with butter in the same pan.
Assemble the sandwiches by spreading sauce on the buns, adding lettuce and tomato, placing the crab cake on top, and finishing with more sauce if you like. Serve warm.