Prepare the Vegetables. Start by removing the outer leaves from the cabbage and cutting it into quarters. Remove the core, then thinly slice the cabbage into strips about 1/4 inch wide. You can use a mandolin or a sharp knife for this.
Prepare the Carrots and Onion. Peel and julienne the carrot into matchstick-sized pieces. Slice the onion into thin strips.
Prepare the Pickling Brine. In a saucepan, combine apple cider vinegar, water, sugar, salt, black peppercorns, mustard seeds, crushed red pepper flakes, and bay leaf. Bring the mixture to a boil over medium heat, stirring occasionally to dissolve the sugar and salt. Once the brine comes to a boil, reduce the heat and let it simmer for about 3-5 minutes to allow the spices to infuse the liquid.
Pack the Vegetables into Jars. While the brine is simmering, place the sliced cabbage, carrots, sliced onion, and garlic cloves into sterilized mason jars. Pack the vegetables tightly but leave some space at the top to allow the brine to fully cover them.
Pour the Brine Over the Vegetables. Once the brine has simmered and the sugar and salt have dissolved, carefully pour the hot brine over the vegetables in the jars. Make sure the cabbage is fully submerged.
Seal and Cool. Seal the jars with lids and allow them to cool at room temperature for 20-30 minutes. Once the jars are cool, transfer them to the refrigerator.
Let It Pickle. Allow the red pickled cabbage to sit in the fridge for at least 4 hours before serving. The flavors will intensify as the cabbage absorbs the brine. For the best results, let the cabbage pickle for 24-48 hours for more robust flavors.