First, press the tofu. Place on some paper towels, then put more paper towels on top. Then put something heavy on top of the tofu. Press for at lest 15 minutes, but the longer the better.
When the tofu is ready, heat the sesame oil on medium high in a non stick pan.
Now, crumble the tofu into the pan. Saute for a minute or two until it starts to get brown. Then add in the chopped onion. Continue to saute until the tofu is brown and the onion is translucent. This will take a 3-4 minutes.
While the tofu is cooking, make the sauce for the tofu. Whisk together the soy sauce, sugar, rice wine vinegar, grated garlic, grated apple, grated ginger and black pepper.
Once the tofu is nice and brown, pour the sauce into the pan. Stir to coat, then cook for another minute or two until the sauce thickens and coats the tofu. Remove from heat.
Then, take two tortillas, sprinkle one side with vegan cheese, then spread half the kimchi around, then spread half the tofu around. Top with more cheese and then another tortilla.
Next, heat the oil for cooking on medium high in a non stick pan, then place the quesadilla in the pan. Brown on one side for 2-3 minutes, then flip and brown the other side. Press down and make sure the vegan cheese has melted.
When done, remove from the pan, and repeat with the remaining ingredients to make the second quesadilla. Cut and serve immediately.