Cut half of the salmon into large chunks, and place into the bowl of a food processor. Add the lemon juice, mayonnaise, and mustard. Process into a paste.: The bowl will take on a faint citrus perfume as the blades transform the chunks into a silky paste. You should see a smooth, cohesive mass clinging to the sides of the processor. This paste adds silkiness to the burgers and helps bind the mixture without overworking the rest of the fish. A clear sign you are done is when the paste holds together and no large pieces remain, but avoid running the processor until it is completely liquefied, which can make the texture mushy. If you process too long, the mixture will feel sticky rather than creamy, so pulse and check often. One mistake to avoid is adding too much liquid at this stage, which will prevent the patties from forming well later.
Dice the remaining salmon and place in a large mixing bowl. Add the green onions, 1/4 cup of breadcrumbs, salt, pepper, and salmon paste from the food processor. Mix gently to combine.: The diced pieces should be tender and slightly glossy, and when combined with the paste they create a pleasing contrast between chunk and cream. Use a spatula to fold the ingredients carefully, so the diced pieces hold their shape. You want even distribution of the green onions and seasoning without turning everything into a uniform mass. The mixture should feel cohesive but not compacted. Overmixing here is the common error, it squeezes out moisture and yields dense patties, so fold gently and stop as soon as the components look combined.
Divide the mixture into four equal mounds and shape into patties. Place the patties on a plate, cover, and chill for 30 minutes (up to 2 hours).: Shaping the mounds with slightly cupped palms gives the patties a gentle dome that will flatten slightly while cooking. Chilling is crucial, it firms the proteins and helps the exterior coating adhere. When you press the chilled patty it should spring back slightly and hold its shape. If you skip chilling, the patties are more likely to fall apart when you try to coat or flip them. A helpful tip is to cover them loosely so condensation does not form on the surface during the chill time.
Spread the remaining 1 cup of breadcrumbs onto a plate. Press both sides of each salmon patty in the breadcrumbs.: The breadcrumb coating should cling evenly to the chilled surface, creating a thin, consistent layer. Press gently but firmly so you do not compress the interior. The texture of the Panko breadcrumbs will give you audible crunch as the crust browns in the skillet. If you see bare spots, press a little more, but avoid overcoating, which can overwhelm the delicate fish. A frequent mistake is wetting the patty before coating, this can cause the crumbs to fall off in the pan.
Heat the butter in a large skillet over medium-high heat. Add the patties and cook for 3-4 minutes until browned. Flip and cook for an additional 3-4 minutes, being sure not to overcook. Transfer to a paper towel-lined plate.: When the unsalted butter melts and just begins to foam, you will smell a nutty aroma that signals the right moment to add the patties. As they sear you should hear a steady sizzle and see the edges take on a warm golden color. Resist the urge to press down with a spatula, this squeezes out juices and can dry the interior. After flipping, watch for gentle bubbling at the edges and an internal color shift to opaque, this indicates doneness. Overcooking is easy to do because salmon goes from juicy to dry quickly, so rely on visual cues and a quick thermometer if you have one.
Serve on buns or a bed of greens with lemon wedges, tartar sauce, and hot sauce.: The final presentation is where you personalize the burger for your table. A squeeze of lemon brightens the crust, while a smear of tartar sauce adds creaminess and a briny counterpoint. If serving on greens, the warm patty resting on crisp leaves creates contrast in temperature and texture. Watch for sogginess if you assemble too early, and consider plating components separately to maintain crispness. One common oversight is piling too many wet toppings on the bun which can make the bread collapse, so serve sauces on the side if you expect leftovers.
