For the tempura: Set a large bowl inside a larger bowl filled with ice and let sit until the top bowl becomes cold to the touch (see Cook's Note). Add the eggs and ice water and whisk vigorously to combine. Sift the flour and cornstarch directly over the liquid and stir only a few times to combine. (It's okay if the batter has some lumps. The gluten in the flour must not be overworked or the batter won't be light.)
Pour the oil into a deep, heavy-bottomed pot to a depth of about 4 inches and heat to 360 degrees F over medium heat. Line a baking sheet with paper towels and get a slotted spoon ready.
Cut a few incisions down the inside length of each shrimp tail so it relaxes, straightens out and no longer curls. Sprinkle with salt.
Dunk the shrimp into the batter to coat. Carefully transfer to the hot oil, gently swirling the oil for even frying on all sides. Fry until the shrimp are light brown with a crispy exterior, 2 to 3 minutes. Remove with a slotted spoon to the paper towels and immediately season with salt.
For the spicy mayonnaise: Stir together the mayonnaise, sriracha, dark soy sauce, lime zest and lime juice in a medium bowl. Taste and adjust the seasoning. Set aside.
For the sushi roll: Completely wrap a bamboo sushi-rolling mat in a tight layer of plastic wrap. (This keeps the rolls from sticking to the mat.)
Put one nori sheet shiny-side down on the wrapped bamboo mat. Gently press and spread about 1/2 cup sushi rice into a shape that matches the nori, leaving a 1/2-inch border around the edges.
Starting at the end closest to you, spoon spicy mayonnaise along the edge of the rice layer. Lay 2 cucumber batons on top and another drizzle of spicy mayo. Place 2 pieces of shrimp tempura on top. Roll the nori up over the fillings, tucking the clean edge underneath to completely encase them. Top with the bamboo mat and gently press, curling your fingers to tighten up the roll.
Remove the roll from the mat and cut into 8 even pieces with a serrated knife, but do not separate them. Shingle the pieces on a plate, slightly overlapping. Top each piece with tobiko wasabi and drizzle the roll with spicy mayonnaise.
Repeat the rolling process with the remaining ingredients. Serve the rolls with freshly grated wasabi, pickled ginger and soy sauce on the side.
For the lightest, crispiest tempura, it's important to keep the batter ingredients very cold.