Peel, then finely grate 1 (1-inch) piece ginger on a Microplane into a medium bowl until you have about 1 tablespoon. Finely grate 3 garlic cloves into the bowl. Add 2 tablespoons mirin, 2 tablespoons soy sauce, 1 tablespoon hon tsuyu, and 1 1/2 teaspoons toasted sesame oil.
Pat 1 1/2 pounds boneless, skinless chicken thighs dry with paper towels. Cut into 2-inch pieces. Add to the bowl of seasonings and massage with your hands until combined. Let marinate at room temperature for 30 minutes.
Add 3 tablespoons all-purpose flour to the chicken and toss with your hands until the flour is absorbed.
Heat 2 cups neutral oil in a large, high-sided skillet over medium-high heat until 330ºF. Meanwhile, fit a wire rack on a baking sheet. Line a second baking sheet or large plate with paper towels. Place 1 cup potato starch in a medium, wide bowl or pie plate.
Coat and fry in batches of 6 to 7 pieces: Add to the potato starch and toss until evenly coated. Place in the hot oil and fry, flipping halfway through, until light golden-brown, 2 to 3 minutes total (the chicken might not be cooked through). Transfer with a slotted spoon or spider to the paper towels.
Heat the oil over medium-high heat to 350ºF. Repeat frying the chicken in batches, flipping halfway through, until darker golden brown and cooked through, about 1 minute per batch. Transfer to the wire rack. Serve with lemon wedges and Kewpie mayonnaise if desired.