Prepare vegan patties according to The Simple Seagan recipe. Cook over medium high heat in a lightly oiled frying pan for 4 minutes on each side, then use a knife or the flat end of a spatula to break the patties up into bite sized strips. Set aside.
To the same pan, add the onions and 1 tsp olive oil and cook over medium high. When the pan starts to look dry, add 1/4 cup vegetable broth and stir the onions. Repeat this process until they are caramelized to your liking or you have used all of the broth. TIP: the less you move the onions around, the better they will cook. When finished, set aside.
Prepare the sauce by whisking together all of the ingredients in a small bowl. Set aside.
Set frying pan over medium heat.
Spread vegan butter over one whole side of each tortilla. Lay tortilla butter-side down in the pan. Over 1/2 of the tortilla, use the back of a spoon to spread 1.5 tbsp of the burger sauce. On the other half of the tortilla, layer vegan cheese, burger patty pieces, a handful of shredded lettuce, and a handful of caramelized onions. Use a spatula to lift the sauced half over the side with filling, then press down on the outside with a spatula. Let this cook for a couple of minutes, pressing down occasionally with the spatula, then flip and cook a couple more minutes on the other side. TIP: When ready to flip, lift the entire quesadilla out of the pan with your spatula. Then with your other hand, turn the pan over the tortilla so it is making contact with the new side you want to cook, and flip both quesadilla and pan at the same time before releasing the quesadilla. This will help prevent filling from falling out.
Repeat process with remaining tortillas and ingredients!