Preparation

Make the teriyaki sauce

• In a small saucepan, combine the soy sauce, sugar, vinegar, sake, scallion, ginger,and garlic. Whisking occasionally over medium-high heat, bring to a boil. In a small bowl, whisk the cornstarch and 1/4 cup water. While whisking the soy sauce mixture constantly, add the cornstarch slurry and cook until bubbling and thickened, about 3-1/2 minutes. Set aside to cool.

Make the burgers

• Put the salmon, egg, scallion, soy sauce, and mayonnaise in a large bowl and gently mix together. Add the panko, quinoa, and 1 tsp. salt, and gently mix until well combined. Divide the mixture into four equal portions, then form into patties the diameter of the bun (about 3/4 inch thick). Cover and refrigerate for 30 minutes or up to 12 hours.

Make the crispy topping

• In a medium bowl, combine the flour, egg, and 1 cup ice-cold water. Mix until just combined; do not overmix. Affix a candy thermometer to the side of a medium saucepan. Add the oil and heat to 350°F. Dribble 1 Tbs. of the batter into the hot oil and cook until golden brown, 3-1/2 to 4 minutes, stirring occasionally and breaking apart any large pieces with a slotted spoon. Transfer to a plate lined with paper towels, and sprinkle lightly with salt if you used all-purpose flour. Repeat until you have about 1 cup of the topping.

Make the toppings and serve the burgers

• Mix the mayonnaise and liquid smoke in a small bowl until well combined. Set aside. Heat 2 Tbs. of the oil in a medium skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until golden brown, 3 to 4 minutes. Season with 1/4 tsp. salt  and pepper. Add 1 tsp. of the butter and the soy sauce to the skillet, and stir to combine.
• In a large nonstick skillet, melt 1/2 Tbs. of the remaining butter over medium heat. Add 2 of the rolls, cut side down, and toast until lightly brown, about 2 minutes. Remove from the skillet and repeat with the remaining 1/2 Tbs. butter and buns.
• Wipe out the skillet, add the remaining 2 Tbs. of oil, and heat until shimmering. Add the salmon patties to the skillet, in batches if necessary, making sure not to crowd the pan. Cook until golden brown, about 3 minutes on the first side and about 2 minutes on the second side. Transfer each burger to a roll bottom. Spoon 1 Tbs. of the teriyaki sauce on each burger. Divide the lettuce, mushrooms, and crispy topping evenly among the burgers. Spread the mayonnaise on the roll tops. Pass the remaining teriyaki sauce at the table.