Marinate the chicken. In a bowl, combine the soy sauce, mirin, rice vinegar, sesame oil, honey, garlic, ginger, and optional chili paste. Add the chicken pieces, toss to coat, cover, and refrigerate for 30 minutes. Keep the raw chicken and its juices away from foods that will be eaten without cooking.
Mix the crepe batter. In a separate bowl, whisk the rice flour, tapioca starch, and salt. Add the eggs, milk, and 1 tablespoon oil and whisk until smooth. Let the batter sit for 10 minutes, then whisk again before using. If necessary, add a tablespoon of water to make it thin enough to spread in the skillet.
Cook the rice crepes. Lightly oil an 8- to 9-inch (20- to 23-cm) nonstick skillet over medium heat. Pour in approximately 1/3 cup of batter and immediately swirl to form a thin round. Cook until the surface is set and the edges release, about 1 to 2 minutes, then carefully flip and cook for another 20 to 30 seconds. Repeat to make at least 4 crepes, stirring the batter each time. Stack the crepes under a clean towel.
Mix the spicy sauce. Stir together prepared mayonnaise, sriracha, and rice vinegar in a clean bowl. Refrigerate until needed. This sauce is made separately and does not contain marinade that touched raw chicken.
Coat the chicken. Remove the chicken from the marinade and discard the used marinade. Drain the pieces well. Place the potato starch in a shallow bowl and toss the chicken in batches until lightly coated. Shake off excess starch.
Fry the karaage. Pour neutral oil into a deep, heavy pot to a depth of approximately 1 to 1 1/2 inches (2.5 to 4 cm), without filling the pot more than one-third full. Heat to 350°F (175°C). Carefully fry the chicken in small batches, turning occasionally, until golden and cooked through, roughly 4 to 6 minutes depending on piece size. Check the thickest pieces with a food thermometer; they must reach 165°F (74°C). Transfer to a rack or paper-towel-lined plate.
Assemble and serve. Lay a crepe flat, spread it with a little spicy sauce, and add one lettuce leaf, a few shiso or mint leaves, and one-quarter of the cooked chicken. Fold in the sides and roll gently. Repeat for the remaining wraps. Serve promptly.