Preheat your grill or stovetop grill pan over medium-high heat.
In a food processor, pulse the salmon fillets until coarsely chopped. Be careful not to over-process; you want a chunky texture.
In a large bowl, combine the chopped salmon, breadcrumbs, mayonnaise, Dijon mustard, lemon juice, garlic powder, salt, black pepper, green onions, and parsley. Mix until just combined.
Form the mixture into 4 equal patties, about 3/4 inch thick. Place them on a plate and refrigerate for 15-30 minutes to help them firm up.
While the patties chill, prepare the Green Goddess dressing by whisking together mayonnaise, sour cream, lemon juice, chives, parsley, anchovy paste (if using), salt, and pepper in a small bowl. Adjust seasoning to taste.
Lightly oil the grill or grill pan. Cook the salmon patties for about 4-5 minutes on each side or until they are cooked through and have nice grill marks.
Toast the hamburger buns on the grill for 1-2 minutes until golden brown.
Assemble the burgers by placing a salmon patty on each bun, adding lettuce and tomato, and drizzling with Green Goddess dressing.