Cut the salmon into small pieces (about 1/4 inch) and place in a medium mixing bowl.
Add the breadcrumbs, chopped cilantro or parsley, eggs, mayonnaise, lemon juice, mustard, 1/2 tsp salt, and 1/2 tsp black pepper to the bowl; gently mix until combined. Let the mixture rest 5–10 minutes to allow the breadcrumbs to absorb moisture.
Divide the mixture into 6 equal portions (about 1/2 cup each) and shape into neat patties.
Heat a little olive oil in a large non-stick grill pan over medium heat. Cook the patties until browned and just cooked through, about 4 minutes per side.
While the patties cook, combine the chopped avocado, red onion, 1/4 cup chopped cilantro, lime juice, 1 tbsp extra virgin olive oil, and salt and pepper to taste in a bowl; gently toss to make the avocado salsa.
Serve the salmon burgers hot with the avocado salsa spooned on top or alongside.