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 oil to 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 chard stems, cook for 3 to 4 minutes, 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.