Combine miso and honey in a bowl. Pour in rice vinegar, followed by hot sauce and salt. Whisk well until mostly smooth. Add chicken halves; cut 3 to 4 slashes into the legs and thighs. Toss chicken in the marinade until well coated. Refrigerate for 2 to 12 hours.
Flip chicken over in the bowl. Sprinkle more kosher salt on top.
Preheat grill to 350 degrees F (175 degrees C) over indirect heat. Add chicken and cover. Grill for 20 minutes and flip chicken over. Cover and grill for another 20 minutes. Brush the excess marinade on top. Cover again and continue cooking until an instant-read thermometer inserted into the center of each thigh reads 165 degrees F (74 degrees C), 5 to 10 minutes more.
Plate chicken alongside lemon wedges sprinkled with cayenne pepper. Squeeze some spicy lemon over the chicken as you cut it up.