Heat oil in a large pan over medium high heat. Add garlic and, just as it starts to turn golden brown, mushrooms. Sautee until the mushrooms start to cook through.
Add chopped chard. You may have to do this in batches, adding more as the previous batch wilts to a manageable amount in the pan. Once all the chard is in and slightly wilted, add broth. Cover and cook until the chard is tender.
Uncover and, if there is still some liquid, allow it to cook almost completely off. When there is just a little bit of liquid, add butter and vinegar. These should combine with the small amount of remaining liquid and thicken into a nice coating for the veggies. Remove from heat. If serving to little ones who can’t manage tortillas, separate out their portion of veggies, chop finely, and mix with some grated cheese.
Wipe out pan and melt a pat of butter, swirling to coat bottom. Place a tortilla in the pan, sprinkle 1/2 of the cheese you want in a single quesadilla, top with veggies, sprinkle rest of cheese, and close with a second tortilla. When bottom layer of cheese melts and bottom quesadilla browns, flip. Melt and brown the other side, and viola! Repeat about 4 more times. Serve with choice of garnish(es).