Finely grate the zest of 1 medium lemon (about 1 tablespoon). Juice the lemon until you have 2 tablespoons. Finely chop 1/2 small bunch fresh dill until you have 1/4 cup plus 2 teaspoons. Coarsely chop 1 medium shallot (about 1/4 cup). Cut 1 1/2 pounds skinless salmon into 1-inch pieces.
Place 1/4 of the salmon (about 6 ounces), lemon zest, 1 tablespoon of the lemon juice, 1 tablespoon of the mayonnaise, 1 tablespoon of the Dijon mustard, 3/4 teaspoon kosher salt, and 1/4 teaspoon black pepper in the bowl of a food processor fitted with the blade attachment. Process until finely chopped and pasty, 10 to 15 seconds. Add the remaining salmon, 1/4 cup of the dill, and shallot. Pulse until the salmon is coarsely chopped, 6 to 8 (1-second) pulses. Do not overprocess.
Transfer the mixture to a large bowl. Add 1/2 cup panko breadcrumbs and stir to combine. Divide the mixture into 4 (about 3/4-cup / about 7-ounce) portions. Shape each portion into a 4-inch wide patty about 1/2-inch thick, and place on a baking sheet. Refrigerate for 20 minutes. Meanwhile, prepare the sauce.
Prepare the following, adding each to the same small bowl as you complete it: Finely grate 1 small garlic clove. Add the remaining 2 teaspoons dill, remaining 1/4 cup mayonnaise, remaining 1 tablespoon lemon juice, remaining 1/2 teaspoon Dijon mustard, a pinch of kosher salt, and a few grinds black pepper. Stir to combine.
Heat 1 tablespoon of the olive oil in a large cast iron or nonstick skillet over medium-high heat until shimmering. Swirl the pan so that the oil coats the bottom of the pan. Using a wide, flat spatula, gently transfer 2 of the patties into the pan. Cook until the bottoms are browned, about 3 minutes. Flip and cook until the second side is browned and the internal temperature is at least 120ºF, 2 to 3 minutes more, being careful not to overcook.
Transfer to a plate. Add the remaining 1 tablespoon olive oil to the pan and repeat cooking the remaining patties.
Split 4 brioche buns. Top each bottom bun with 1 salmon burger patty. Divide the sauce over the burgers and top with a small handful of baby arugula. Close with the top buns.