Place cubed salmon in the bowl of a food processor fitted with blade attachment. Pulse until the fish is pretty finely chopped, but not a paste. Transfer to a large mixing bowl.
To the salmon, add sour cream, chives, tarragon, fresh dill, chopped capers, Dijon mustard, fresh lemon juice, and the zest of lemon. Season with Kosher salt and freshly ground black pepper. Use a spatula or your hands to gently combine all the ingredients.
Divide the salmon mixture into 6 equal portions, then use your hands to shape into uniform patties. If not cooking immediately, cover with plastic wrap, then refrigerate for up to 8 hours.
Place mayo, chopped tarragon, and white wine vinegar in a small bowl, then stir well. Season with Kosher salt, taste, and adjust seasoning as needed. Set aside.
Preheat a nonstick skillet (or grill) over medium-high heat, then coat with nonstick cooking spray. When hot, add salmon burgers and cook.
Place a generous amount of tarragon mayo on both sides of the burger bun, then place a heaping mound of salad greens on the bottom bun. Place a cooked salmon burger patty on top, along with any burger toppings you like.