Heat cast-iron griddle on medium-low over 2 burners. Meanwhile, thinly slice 1 jalapeño half into half-moons and finely grate remaining half; keep separated.
In medium bowl, toss carrots, daikon, and sliced jalapeño with lime juice and 2 tsp fish sauce; let sit 10 minutes.
Meanwhile, in large bowl, stir together scallion whites, garlic, ginger, lime zest, grated jalapeño, half of cilantro, and remaining 2 tsp fish sauce. Stir in chicken just until combined. Spread mixture on top of 4 tortillas (about ⅓ cup on each), leaving ⅛-inch border.
Brush griddle with oil. Cook tacos, chicken sides down, until chicken is browned and cooked through, 2 to 3 min. Flip and cook until tortillas are golden brown, 1 to 2 minutes. Meanwhile, spread chicken on remaining 4 tortillas, then cook after first batch is done.
Serve tacos topped with pickled vegetables, cucumber, and remaining cilantro and scallion greens alongside lime wedges if desired.
