Add the miso paste, sake, mirin and brown sugar to a medium saucepan. Heat over medium low heat, until the sugar dissolves and the mixture is simmering, about 5 minutes. Remove from heat and let the marinade cool completely.
Add the steak and the marinade to a resealable plastic bag, making sure the steak has been covered all over. Refrigerate for at least 1 hour and up to 24 hours.
Heat a gas grill to medium-high heat. Remove the steak from the marinade, wiping off any excess.
Add the ribeyes to the grill and sear, for 4 minutes. Keep an eye out for flare-ups. If a flare-up occurs and lasts for more than a few seconds, move the steaks out of the fire to prevent overcooking the steak. Flip and cook the steaks until medium-rare, about an additional 4 minutes. The internal temperature of the steak should be 130ºF (54ºC).
Remove from the grill and let the meat rest on a cutting board for about 15 minutes.
Rub the bok choy with oil and add the bok choy to the grill and cook, until the bok choy is slightly browned, about 5 minutes. Remove and cut trim the bottoms off the bok choy to separate the individual leaves. Set aside.
Thinly slice the steak, discarding any large pieces of fat.
Divide rice among serving bowls. Top with slices of steak and bok choy and finish with sesame seeds.