To cook the rice: Rinse the rice until the water runs mostly clear. Cook in a rice cooker using the rice setting. Or combine the rice and water in a pot. Bring to a boil over medium-high heat, reduce to low, cover, and simmer for 15 minutes. Remove from the heat and let rest, covered, for 15 minutes. Let the rice cool until just slightly warm.
To make the spicy mayo: Whisk together the Japanese mayonnaise, Sriracha, and rice vinegar until smooth. Reserve half of the sauce for serving.
To prepare the shrimp: Season the shrimp with the salt and white pepper. Toss with half of the spicy mayo until evenly coated. Coat each shrimp with the panko breadcrumbs.
To assemble the sushi cups: Preheat the oven to 400°F (204°C). Layer two halved pieces of roasted seaweed in each cup of a standard 12-cup muffin tin. Add about 1 tablespoon of cooked rice and gently press it into the bottom. Place one breaded shrimp on top of the rice in each cup.
To bake: Bake for 15 minutes, or until the shrimp are cooked through and the breadcrumbs are lightly golden.
To finish and serve: Top each sushi cup with the toasted sesame seeds, diced cucumber, and diced avocado. Drizzle with the reserved spicy mayo or serve it on the side. Serve immediately.