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Heat butter over medium heat in a large heavy-bottomed frying pan. Add bell peppers and onion and cook, stirring occasionally until vegetables are soft, 7-8 minutes. Transfer to a plate and set aside.
Cut three-quarters of the salmon into ¼-inch dice and place in a medium bowl. Cut remaining salmon into 1-inch pieces and add to a food processor with mustard, 1 tablespoon mayonnaise, the lemon juice, and lemon zest. Pulse ingredients into a paste, then transfer to bowl containing the 1/4-inch salmon pieces.
Add 2 tablespoons of panko breadcrumbs, the cooked onion mixture, and Cajun seasoning; mix gently until combined. Form 4 equal-size salmon patties about 4 inches wide, using a 4-inch metal ring if you like for even shaping. Pour remaining panko onto a plate. Turn patties in panko to coat. Place on a plate or a sheet pan lined with parchment paper. Refrigerate patties at least 30 minutes or up to 1 hour.
Meanwhile, prepare spicy mayo: Add ½ cup mayonnaise, the hot sauce, lime juice, and cayenne to a small bowl and mix until combined. Place in refrigerator until ready to serve (or up to 1 week ahead).
Heat the large frying pan over medium-high heat. Add 1 tablespoon grapeseed oil. Transfer 2 patties to hot pan. Cook patties until browned on the bottom, 4-5 minutes. Flip patties and continue cooking until second side is browned, about 4 more minutes. Transfer to a plate and tent with foil. Wipe out pan with a paper towel to remove crumbs. Repeat to cook remaining patties with remaining oil.
Spread spicy mayo on cut sides of hamburger buns. Arrange tomato slices, arugula, and pickles on each bottom bun. Top each with a salmon patty and the bun top, and then serve.