Start by removing the skin from the salmon and finely chopping it into 1/4-inch pieces—this small dice is crucial for the right texture. Divide the chopped salmon in half, setting one half aside on a cutting board. In a food processor, combine the other half of the salmon with the Greek yogurt, Dijon mustard, and lemon zest. Pulse until you have a smooth paste with no large chunks; this paste acts as a binder and adds richness to the burgers. Transfer the paste to a large mixing bowl.

Add the reserved finely chopped salmon to the paste in the bowl along with the sliced green onions, fresh dill, drained capers, Panko breadcrumbs, smoked paprika, salt, and freshly ground black pepper. Using a gentle folding motion with a spatula or your hands, combine everything just until the ingredients are evenly distributed—avoid overmixing, which can make the burgers dense and tough. I like to add the lemon juice at this stage rather than earlier; the acid helps brighten flavors without being processed away. The mixture should hold together but still have a slightly loose, textured consistency.

Divide the burger mixture into four equal portions and gently shape each into a patty about 3/4-inch thick. Handle the mixture as little as possible to keep the burgers tender. Place the formed patties on a parchment-lined plate or directly on a clean work surface. Let them rest in the refrigerator for at least 15 minutes if you have time—this helps them hold together during cooking—though you can proceed immediately if needed. While the patties chill or just before cooking, lightly toast the brioche buns in a dry skillet or toaster until they're golden and have a slight crunch on the interior.

Heat your grill or grill pan to medium heat (around 375-400°F if using a grill thermometer). Lightly brush or add the 2 tablespoons of olive oil to the grates or pan surface. Carefully place the patties on the hot grill and resist the urge to move them—let them cook undisturbed for 4 to 5 minutes until the bottom develops a light golden crust. Gently flip once and cook for another 4 to 5 minutes on the second side until cooked through. The burgers are done when they feel firm to the touch and an instant-read thermometer inserted into the center reads 145°F. I find that not moving the patties frequently is key to getting a nice crust while keeping the interior moist.

Remove the cooked burgers from the grill and let them rest for 2 to 3 minutes—this allows the juices to redistribute and keeps them moist when you bite into them. Place a burger patty on the bottom half of each toasted brioche bun. Layer with butter lettuce leaves and a generous dollop of tartar sauce on top, then crown with the bun's top half. Serve immediately while the burgers are still warm and the buns retain their crunch.