Cut the Salmon Fillet - If still attached, remove the skin from the fish. Cut the salmon fillet into 1-inch pieces. Trim off and discard any grey areas from beneath the fillet.
Make the Patties - In a food processor or blender, add salmon cubes, dijon mustard, shallots, capers, dill, salt, and pepper. Pulse and scrape down sides until coarsely ground. There should be some small chunks and not overly pureed.
Make a Panade - Transfer the salmon mixture to a medium-sized bowl. Add the breadcrumbs, mix with a spatula until combined.
Form the Patties - Evenly divide the salmon mixture into four portions, about 7 ounces each. Shape into 4-inch wide patties, about 3/4-inch thick. Lightly brush each side of the patties with olive oil. Cover and refrigerate until ready to cook.
Lemon Dill Sauce
Make the Lemon Dill Sauce - In a small bowl, combine the mayonnaise, mustard, shallots, basil, dill, lemon zest, lemon juice, and cayenne pepper.
Cook the Burgers
Cook on the Stovetop -Heat a grill pan, large cast iron, or nonstick skillet over medium-high heat. Evenly grease the surface of the pan with about 1 tablespoon of olive oil. Once the oil is hot, add patties. Cook until golden brown, about 3 to 4 minutes. Flip and cook until the internal temperature reaches about 130ºF (54ºC), about 2 to 4 minutes.
Cook on the Barbecue -Heat grill over medium heat. Clean and oil the grill grates. Once hot, add the patties. Cover and cook until char marks appear, about 3 to 4 minutes. Flip, cover, and cook until the internal temperature reaches about 130ºF (54ºC), about 2 to 4 minutes.
Toast the Buns - Right before serving, grill the buns cut-side down. Heat until toasted, about 2 to 3 minutes.
Add Burger Toppings - Add about 1 tablespoon of lemon dill sauce to each toasted bottom bun. Top with a salmon patty, avocado, tomato, onion, and lettuce.
Notes
Make-ahead: The salmon patties can be formed and stored in an airtight container in the refrigerator 2 days before cooking.
Storing: Cover and store cooked patties for up to 3 days in the refrigerator. Refrigerate the lemon dill sauce for up to 3 days.
Reheating: Reheat in the microwave on high power in 30-second intervals until warmed through.