In a large bowl, combine the miso paste, beer, soy sauce, sugar, sesame oil and cayenne pepper. Stir until the miso and sugar are completely dissolved. Set aside 1/2 cup of the sauce for basting during grilling. Submerge the chicken in the remaining marinade, cover bowl and refrigerate for at least 2 hours.
Preheat barbecue for medium-high heat.
Lightly oil the barbecue grate. Remove the chicken from the marinade and discard marinade.
Barbecue chicken for 6 to 8 minutes per side basting during the last few minutes with the reserved sauce. The chicken is done when its juices run clear.