Using a sharp knife, finely chop the salmon by hand (Note: You can freeze the salmon for about 10 minutes to make the process a bit easier.)
Place the chopped salmon in a large bowl. Add the scallions, mayonnaise, shallot, dill, and lemon zest and stir to combine. Divide into 6 portions.
Transfer to the refrigerator to chill and firm slightly, about 30 minutes (Note: The mixture will be relatively loose. This will help them keep their shape during cooking.)
Liberally coat a large cast iron skillet with canola oil and place over medium-high heat.
Pour the breadcrumbs into a large, shallow dish. Working one portion at a time, gently press into a patty about 1-inch thick.
Dredge liberally in the breadcrumbs and shake off the excess. Without moving in the pan, sear until the bottom is deep golden brown, about 5 minutes.
Flip, and cook the other side until the inside is cooked through, about 7 additional minutes.
In a small bowl, combine the yogurt, mayonnaise, dill, vinegar, and salt. Whisk to combine and set aside.
Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Peel the potatoes and halve lengthwise. Cut each half into four pieces so each potato yields eight wedges and place in a large pot. Cover with water by 3 inches and season very liberally with kosher salt. Bring to a boil, then cook for 5 minutes.
Remove from the water with a spider or slotted spoon and transfer to a paper towel lined plate.
Transfer the blanched potatoes to the lined baking sheet. Bake until the tops are golden brown, about 30 minutes. Flip and cook until the other side is cooked through and the insides are tender, about 30 additional minutes.
Serve the salmon burgers warm on brioche buns with sauce, lettuce, and fries on the side.