Remove the skin from the salmon filet and cut it into ¼ inch cubes. Take ½ of the salmon and mince it into very small pieces and place the meat back into the bowl.
Thinly slice half of the purple onion, sweet peppers, and green onions. Dice the other half of the purple onion into small pieces and set aside.
Add seasonings, ½ of the chopped green onions, and ½ of the flour into the bowl of salmon and mix until combined. Once combined, add the remainder of the flour, mayo, and egg. Mix until all ingredients are incorporated.
In a medium saucepan, add the vinegar and allow it to come to a boil. Once the bubbles start to form, add in the thinly sliced onions and peppers. Simmer for 2 minutes and remove from heat. Transfer into a heat-safe bowl and refrigerate.
In a large frying pan, add butter and olive oil over medium heat. Add in ¼ cup scoops of the salmon mixture and cook for about 2 minutes on each side, or until golden.
Butter the brioche buns and lay them face down in a saucepan and toast until golden.
Assemble the burgers with the quick pickled veggies, lettuce, and your favorite sauce. Enjoy!