Slice the stems into 1-inch pieces and reserve. Stack the chard leaves into a pile. Roll together into a bundle and slice into 1/2-inch ribbons.
Heat the oil in a Dutch oven over medium heat. Add the bacon and saute until browned, rendering the fat. Add the onion and garlic and cook until translucent. Add the red pepper flakes and the chard stems, cook for 3 to 4 minutes and then season with salt and pepper.
Begin to add the chard ribbons in batches. Once the chard wilts down, add the next batch. Stir occasionally until completely tender, about 5 minutes. Add the balsamic vinegar and serve.