In a bowl, mix together the soy sauce, brown sugar, oyster sauce, rice wine vinegar, sesame oil, garlic paste, ginger paste, red pepper flakes, and water (if using).
Pour the sauce over the steak and vegetables, stirring to combine. Cook for an additional 2-3 minutes until the sauce thickens slightly.
Remove the mixture from heat and set aside.
On a griddle, place a large flour tortilla and sprinkle a generous amount of shredded cheese on one half.
Add a portion of the steak and vegetable mixture on top of the cheese, then fold the tortilla over.
Cook until the tortilla is golden brown and the cheese is melted, flipping once.
Repeat with remaining tortillas and filling.
Serve with chipotle mayonnaise or sour cream on the side.
Notes
Feel free to adjust the spice level by adding more or less red pepper flakes.
For a vegetarian option, substitute the steak with grilled mushrooms or tofu.
These quesadillas can be made ahead of time and reheated in the oven.