For the asparagus: Slice each asparagus spear in half lengthwise. Wrap each piece of asparagus in a 4-by-1 3/4-inch sheet of nori and set aside.
For the marinated mushroom: Trim all sides of the mushroom until you have a 4-by-1-by-1-inch rectangular block. Slice the block lengthwise into 4 thin slices. In a small mixing bowl, whisk the soy sauce and sugar together until the sugar dissolves.
Place a small nonstick skillet over medium heat. Add the oil and mushrooms and saute for 2 to 3 minutes. Pour the soy mixture over the mushrooms and toss together until fully and evenly coated. Transfer the mushrooms to a small plate and let cool. Wipe out the skillet and reserve for cooking the tamago.
Once the mushrooms have cooled, arrange the pieces onto half of a 4-inch square sheet of nori. Fold the remaining part of the nori over the mushrooms to wrap.
For the cucumber: Trim the ends of the Persian cucumber until 4 inches long. Set aside.
For the tamago (Japanese omelet): Beat together the eggs, soy sauce and pepper in a bowl. Place the reserved skillet from the mushrooms over medium-low heat. Add 1/2 teaspoon of the oil and one-third of the egg mixture. Gently scramble with a spatula until the eggs are barely set. Push the eggs to the side of the skillet, towards you. Add another 1/2 teaspoon oil to the empty portion of the skillet, add another third of the egg mixture and allow the eggs to cook into a flat disc. Once the eggs are barely set, flip the previously scrambled eggs over the flat disc of eggs and continue to flip until a roll forms. Push the roll to the side of the skillet again. Pour the remaining oil and egg mixture into the skillet and repeat the last step until you have a thicker, tight omelet roll. Slide the tamago onto a cutting board and cool. Once cool enough to handle, trim the omelet down to a 4-by-1/2-1/2-inch rectangle (scraps can be snacked on!). Set aside.
For the sushi rice: Combine the vinegar, sugar and salt in a small mixing bowl and whisk together. Sprinkle two-thirds of the vinegar mixture over the cooked white rice and gently fold together until fully incorporated.
Place one-quarter of the sushi rice into a mixing bowl and fold in the tobiko. Set aside. Sprinkle the remaining vinegar mixture over the cooked green pearl rice and gently fold together until fully incorporated. Set aside.
For the palm tree: Place one 4-by-7-inch sheet of nori onto a bamboo sushi rolling mat. Press a thin layer of plain rice over the center 3 inches of nori. Place the nori-wrapped mushrooms onto the center of the rice standing up and build a thin layer of rice next to and on top of the mushrooms to keep them balanced.
Gently flatten the rice on top of the mushroom and place 2 wrapped asparagus spears side by side, cut-side down, on top. Top the asparagus spears with a very thin layer of rice and top with the remaining 2 asparagus spears, cut-side down. Fill the remaining surface of nori that the tree is sitting on with rice. Add a 4-by-2-inch sheet of nori on one end to create a tail. Using the sushi mat, with the tail end away from you, carefully wrap the rice-topped nori over the palm tree and roll completely until sealed. Slice the roll into 6 pieces and transfer to a platter.
For the geometric pattern: Place a 4-by-5 1/2-inch sheet of nori onto the sushi mat and top with a thin layer of tobiko rice. Lay the cucumber across the center of the tobiko rice. Then, using the sushi mat, wrap the rice and nori around the cucumber to seal. Slice the roll in half lengthwise. Turn each half cut-side down and slice in half lengthwise so the entire roll is quartered. Set aside.
Place a 4-by-7-inch sheet of nori onto the sushi mat and top with 2 of the quarter rolls on the bottom edge of the sheet, side by side, so the top creates a V shape. Nestle the bottom half of the tamago into the V-shaped opening and set the remaining 2 quarter rolls against the top half. Using the sushi mat, carefully wrap the nori around the cut rolls until sealed.
Place a 4-by-7-inch sheet of nori onto the sushi mat and top with a thin layer of the green pearl rice. Place the previously assembled sushi block in the center of the green rice. Using the sushi mat, carefully wrap and roll the green rice-topped nori around the sushi block to seal. Slice the roll into 6 pieces and transfer to the platter. Serve.