Make the remoulade. Mix all of the ingredients together in a bowl and refrigerate until ready to use. Discard the garlic clove just before using.
Being careful not to overwork the meat, form the chuck into 4 patties. Season each with salt and pepper. Make an indentation on each patty with your thumb.
For the crabs, mix the flour, paprika, garlic powder, salt and pepper on a plate. Dredge each crab in the flour mixture. Heat butter in a skillet, place the crabs in the pan, top shell side down. Flip each crab after 3 minutes and cook the other side for an additional 3 minutes or until the crabs are red.
Cook the burgers. Heat a cast iron pan until very hot. Add a pat of butter and then the patties, indentation side down. Flip after 3 minutes and cook until desired doneness (I like medium rare).
When the burgers are done cooking, transfer them to a plate and place the buns in the pan cut side down for a light toasting. Assemble the burgers. Bun, remoulade, arugula, burger, tomato, crab, more remoulade if wanted, top bun.
Enjoy the ENTIRE thing!