Preparing the ginger beef.
Peel and grate the gingerroot in a circular motion until you have about 3 tablespoons of ginger. Squeeze the grated gingerroot to release all the juices. You should have about 2 tablespoons ginger juice for the marinade. Discard the ginger fiber.
Stir the ginger juice, soy sauce, and mirin together in the bottom of a small dish. Put the meat in the dish, turning to coat both sides with ginger marinade. Refrigerate the meat for 30 minutes, turning it once or twice.
Pour the vegetable oil into a heavy frying pan and place it over high heat. Once hot, fry the marinated meat, turning it over once or twice, for 4 to 5 minutes. Reduce the heat to low and pour in the ginger marinade. Cook the meat 4 to 5 minutes more, turning 2 or 3 times, until well done. Take the meat out of the pan and let it cool for 5 to 10 minutes before slicing it, allowing time for the juices to settle down. Slice the meat into 1/4-inch-thick strips. Set aside briefly or refrigerate overnight.
Assembling the Roll.
Bring a large pot of water to a boil. Drop in 1 or 2 lettuce leaves at a time, blanching them for 3 to 5 seconds at most. Using tongs, remove them from the water and then plunge them into a bowl of ice water to stop the cooking and set their color. Drain them on paper towels. Repeat until all the leaves are blanched. Slice off, without slicing through the lettuce, the raised back of the lettuce ribs.
Place the plastic-wrapped bamboo mat in front of you with its slats parallel to the edge of the table or countertop. Lay 2 blanched lettuce leaves down, ribbed side up, on the edge of the wrapped bamboo mat closest to you. The stalk ends should be slightly overlapping in the middle. Carefully open them up until they're completely flat. Their combined shape should be roughly 7 inches wide x 8 inches long. Fold in the tips of the lettuce if they extend off the bamboo mat. If a leaf tears, replace it or patch it with pieces of an extra leaf.
Spread 1 cup of prepared sushi rice evenly on the lettuce, leaving a 2-1/2-inch-wide strip of lettuce not covered with rice on the edge farthest away from you.
Lay 2 pieces of pickled mountain burdock, end to end, across the center of the rice, spanning the length of the lettuce. Bunch one-fourth of the ginger beef up next to the pickled burdock, spanning the length of the lettuce. Roll up this lettuce-wrapped roll as you would a nori-wrapped sliced roll: Lift the edge of the mat closest to you, keeping your fingers over the fillings. Roll the mat and its contents away from you until the edge of the mat touches straight down onto the far edge of the rice, enclosing the fillings completely. Tug on the mat to tighten the roll. Lift the edge of the mat up and finish rolling it up. Tug on the mat again to tighten the roll.
Lay a piece of plastic wrap over the roll. Cut the roll into 8 equal pieces, wiping your knife with a damp towel before each slice. Make 3 more rolls, following these instructions. Serve immediately or set aside in a cool place, covered, up to 2 hours.