Prepare the salmon patties. Cut the salmon into bite-size pieces and pulse in a food processor until coarsely chopped. Alternatively, chop the salmon by hand. Combine the salmon with the remaining ingredients including cilantro, shallots, yogurt, egg, lime, chili powder, smoked paprika, cumin, brown sugar, garlic, salt, pepper, and oregano. Let the salmon mixture sit for 20 minutes.
Meanwhile, make the avocado crema. Blend together the avocado with yogurt and juice of ½ lime. Season with salt and pepper. Set aside.
Prepare the crispy onions. Combine the thinly sliced onion with buttermilk and let it sit for 10 minutes. Preheat frying oil to 375 degrees F.
Season the flour with salt and dredge the onions in the flour mixture, shaking off the excess. Carefully drop the onion into the hot oil and fry until golden brown. Remove the onions with a slotted spoon and drain the excess oil on paper towels. Season the onions with salt.
Shape the salmon into 4 patties. Cook the salmon on the grill or in a saute pan over medium-high heat until golden brown on both sides.
Assemble the burgers. Top the bottom brioche bun with butter lettuce leaves, the salmon patty, avocado crema, crumbled feta, and a pile of crispy onions. Top with the remaining bun and serve immediately.
Notes
You can prepare the avocado crema and salmon patties the day before.