Process egg, lime zest, lime juice, mustard, ginger, soy, sugar and 3 tablespoons canola oil for a few seconds in the food processor. With motor running, slowly add – in a constant 1/8-inch-thick stream – the remaining cup of canola oil until it is all used up and mayonnaise consistency is reached.
Transfer to a serving bowl and keep refrigerated. (The addition of ginger and soy sauce make this mayonnaise a bit thinner than usual.) Clean the bowl and blade of the processor.
In a small skillet over medium heat, cook carrot, onion, and mushrooms in 1/2 cup water, uncovered, stirring frequently, until carrot and onion are softened, about 4 minutes (the liquid will have evaporated and the vegetables will begin to brown).
Cool slightly. Combine shrimp, fish, vegetable mixture, yolks, sea salt, soy sauce, and sugar in the bowl of a food processor and pulse a few times till mixture forms a chunky paste. Scrape into a bowl and chill, covered, for 30 minutes.
Heat the 4 tablespoons canola oil in a large nonstick skillet over moderate heat until shimmering. While oil is heating, form shrimp mixture into eight 1/2-inch thick patties. I used a ring mold (3.5 inches) and filled it halfway.
Fry cakes until a light golden, about 2 minutes on each side, then transfer with a spatula to paper towels to drain. Serve on a bed of greens — I prefer a peppery variety such as arugula or mizuna — with a dollop of the mayonnaise; garnish with chives, and pass remaining mayonnaise on the side.