Make the spicy mayo: In a small bowl, stir together the Kewpie spicy mayo, lime juice, smoked paprika, sweet chili sauce, a pinch of flaky sea salt, and cracked black pepper until smooth. Set aside.
Make the slaw: Add the green cabbage, purple cabbage, red onion, and scallions to a medium bowl. Toss with ⅓ cup of the spicy mayo, then season with sea salt and pepper to taste. Refrigerate while you prepare the shrimp.
Prep the breading station: Add the flour, baking powder, and a pinch of smoked paprika to a shallow bowl and whisk to combine. In a second bowl, whisk together the eggs and milk.
Bread the shrimp: Pat the shrimp dry with paper towels. Working in batches, coat each shrimp in the flour mixture, dip into the egg mixture, then return it to the flour and press lightly so the coating sticks.
Fry the shrimp: Add about 1 to 2 inches of frying oil to a heavy-bottomed skillet or Dutch oven and heat to 350°F. Fry the shrimp in batches for about 2 to 3 minutes, turning as needed, until golden brown and cooked through. Transfer to a wire rack or paper towel-lined plate and immediately season with smoked paprika and flaky sea salt.
Toast the brioche: Butter the cut sides of the brioche and toast in a skillet over medium heat until golden.
Assemble the sandwiches: Spread additional spicy mayo on the toasted brioche, add a generous layer of slaw, and pile the crispy shrimp on top. Finish with the top bun and serve right away.