To prepare the salmon, cut all of the salmon into small, bite-sized chunks. Place half of the cut salmon into a food processor and pulse until coarsely ground. Transfer both the chopped and ground salmon into a large mixing bowl.

To the large mixing bowl, add the fresh dill, green onions, Greek yogurt, lemon juice, salt, and freshly ground black pepper. Gently mix until just combined.

Divide the salmon mixture into four equally sized portions (about 6 ounces each). With damp hands, firmly form each portion into a 1-inch thick patty. Chill the patties in the refrigerator for at least 10 minutes to help firm them up.

While the salmon patties are in the refrigerator, make the sauce. In a small bowl, whisk together the Greek yogurt, fresh dill, lemon juice, lemon zest, salt, and black pepper until the sauce is smooth. Cover and place in the refrigerator until ready to use.

To cook the salmon patties, heat a large skillet or grill pan over medium heat and add the olive oil. If using a grill pan, brush the pan with the olive oil. Once hot, place the salmon patties in the pan and cook for 4-5 minutes on each side, or until they are golden brown and cooked through. The internal temperature of the salmon patties should reach (145°F or 62°C) when done.

Toast the brioche buns under a broiler or on the same pan if desired.

To assemble the burgers, spread a generous amount of the dill sauce on the bottom halves of each bun. Add a salmon patty to each, and if using, top with sliced red onion, tomato and arugula. Finish with the top half of the bun. Serve and enjoy!