In a small food processor (or chopping by hand), pulse the ahi tuna until it’s coarsely ground (you can also ask your fish monger to do this for you). Transfer to a medium mixing bowl.
To the ground tuna steaks, add 2 tablespoons of Dijon mustard, 2 tablespoons chopped parsley, the shallot and ½ teaspoon sea salt. Mix until completely incorporated.
Form the fresh tuna mixture into 4 equal balls and pat into 1-inch thick burger patties about the size of your palm. Set aside on a plate and transfer the patties to the fridge to chill for at least 15 minutes.
Meanwhile, make the fennel slaw: in a medium mixing bowl, toss the fennel with 2 tablespoons lemon juice, 1 tablespoon oil and ½ teaspoon sea salt. Toss until well-coated and set aside, stirring every 10 minutes or so to redistribute, until the fennel is softened.
Make the aioli: In another small mixing bowl, combine the mayonnaise with the remaining 2 tablespoons Dijon mustard, 2 tablespoons parsley, 2 tablespoons lemon juice, capers, garlic and ¼ teaspoon salt.
Cook the ahi tuna steak burgers: place a large nonstick or cast iron skillet over a medium-high flame and coat with a thin layer of avocado oil. When the pan is hot, add the ahi tuna burgers. Cook the patties until nicely browned on the first side, 2 to 3 minutes.
Make sure not to flip the burgers too early—they need to be nicely browned on the first side in order to turn without breaking. Carefully flip using two flat spatulas and cook for another 1 to 2 minutes, until opaque up the sides for medium.
To serve, slather the toasted buns with the caper aioli. Top the bottom half of each bun with a tuna burger patty, and divide the fennel slaw and arugula between them. Enjoy immediately with the remaining aioli on the side!
