Add soy sauce, mirin, sake, garlic, brown sugar, ginger, and Thai chili to the jar of a blender. Blend on high until smooth. Reserve 2 tablespoons marinade; pour remaining marinade into a resealable plastic bag.

Add steak to the resealable bag, close the bag, and marinate in the refrigerator for at least 30 minutes or up to overnight.

Heat oil in a large skillet over medium heat. Remove steak from the marinade, shake off excess, and cook and stir steak in the skillet until browned, about 5 minutes. Discard the remaining marinade.

Meanwhile, add broccoli to a microwave-safe steamer, and steam in the microwave until tender, about 3 minutes. Roughly chop the broccoli.

Remove steak to a plate to keep warm, and drizzle with the 2 tablespoons reserved marinade. Rinse the skillet and dry with a paper towel. Place 1 tortilla in the skillet, and add a 1/8 portion each of cheese, steak pieces, broccoli, and bell pepper to the tortilla. Fold tortilla in half. Press down with a spatula, and cook until the tortilla begins to brown and cheese melts, about 3 minutes. Carefully turn the quesadilla over and cook until browned on the other side, about 3 minutes more. Repeat with remaining ingredients.