For the shrimp cakes: Heat the vegetable oil in a 12-inch nonstick skillet over medium-high heat. Add the mushrooms, shallots and carrots. Cook, stirring frequently, until the carrots begins to soften, 6 to 7 minutes. Transfer the mixture to a food processor. Add the shrimp, breadcrumbs, egg, lemon zest, salt and pepper. Pulse until combined but still chunky. Form the mixture into 15 patties, each 2 inches in diameter and 1/2-inch thick, using damp hands. Place on a plastic wrap-lined baking sheet and refrigerate for 30 minutes.
For the ginger butter: Mix together the butter, ginger and soy sauce in a medium bowl, using a rubber spatula, until combined. Place the mixture on a piece of plastic wrap. Fold the plastic wrap over and form the mixture into a log about 7 inches long and 1-inch in diameter. Refrigerate until firm, at least 30 minutes. When ready to serve, slice the butter into 1/4-inch-thick pieces.
Heat the vegetable oil in a 12-inch nonstick skillet over medium-high heat. Add half of the patties and cook until golden, 3 to 4 minutes per side. Cook the remaining patties, adding extra oil if needed. Serve the shrimp cakes while still warm with a slice of ginger butter on top.
Use any leftover ginger butter to top chicken or fish.