Cook the salmon. Preheat an oven to 375°F (190°C). Place salmon on a parchment-lined baking sheet. Pat the salmon lightly with a paper towel, then season with Kosher salt and black pepper. Dollop with two tablespoons of butter, then bake until an internal temperature of 120°F (49°C) is reached, about 12-15 minutes, based on thickness.
Sauté the shallots. While the salmon is cooking, heat olive oil in a small skillet over medium heat. Add shallot and cook until translucent, about 2 minutes, stirring often. Set aside.
Flake the salmon. Allow salmon to cool for about 5 minutes, then flake with your hands or a fork, discarding the skin.
Mix ingredients for salmon patties. In a large bowl, combine flaked salmon, sautéed shallots, and remaining salmon patty ingredients. Mix lightly until well incorporated, trying not to break the salmon pieces down too much.
Portion the salmon cakes. Use a large cookie scoop to equally divide the salmon mixture. For a ~2 lb salmon fillet, you should get 10 patties. Use your hands to gently press down and form into uniform, round, flat cakes. (It helps to wear kitchen gloves or slightly wet your hands first.)
Cook on a stovetop. Preheat remaining 1 Tbsp oil and 1 Tbsp butter in a large nonstick skillet over medium heat. Cook 5 patties at a time, for 2-3 minutes per side, until light golden brown on both sides. Transfer to a plate, then repeat with remaining salmon cakes.
Serve immediately. Garnish with capers, lemon wedges, marinated or pickled red onions, fresh herbs, micro greens, and Greek yogurt sauce.
Make the yogurt sauce: Combine all sauce ingredients together in a small bowl, then whisk to combine.