How to Make It
Heat 2 cups canola oil in a stockpot over medium-high to 350°F.
Toss shallot rings with rice flour; shake off excess flour. Carefully drop dusted shallots into hot oil, and stir. Cook until evenly browned, 3 to 5 minutes. Remove with a fine-mesh strainer, and drain on paper towels. Season with ½ teaspoon of the salt.
Pulse fish fillets, edamame, carrot, lemon zest and juice, olive oil, and hot sauce in a food processor until combined, 7 to 8 times. Season with pepper and remaining 1 teaspoon salt; pulse again until everything comes together, about 5 times. (The mixture should have an uneven texture but come together into a ball.)
Stir together mayonnaise and yuzu juice, and set aside.
Rub a grill grate or a grill pan with canola oil, and heat to medium-high (400°F to 450°F). Shape fish mixture into 4 (6-ounce) patties; cook until a thermometer inserted in center of patties registers 140°F, 3 to 4 minutes per side.
Spread both halves of rolls with yuzu mayo. Layer lettuce, tomato slices, crispy shallots, and fish burgers between roll halves. Serve immediately.