Whisk hoisin sauce, brown sugar, rice vinegar, garlic, chili sauce, and ginger together in a bowl until marinade is smooth; pour into a resealable plastic bag. Add skirt steak, coat with marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator for at least 24 hours.
Remove steak from the marinade and shake off excess. Discard remaining marinade.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Cook steak until it starts to firm and is reddish-pink and juicy in the center, 6 to 8 minutes per side. Sprinkle blue cheese over the steak and cook until cheese just starts to melt, 2 to 3 minutes. An instant-read thermometer inserted into the center of the steak should read 130 degrees F (54 degrees C).