Cut the cabbage in half, cut out the core, cut it into quarters, and then in 2-inch pieces.
Peel and trim the end of the beet, then cut it into 1/2-inch half rounds.
Layer the cabbage, beets, garlic, and dill tightly into a large pot or a container (big enough to fit a small plate).
Bring water, vinegar, sugar, and salt to a boil. Once it boils, remove the pot from heat, add the avocado oil, mix to combine and pour the mixture into the container with veggies.
Add a plate to cover the veggies, then top with a can or something heavy. Let it sit overnight on the counter. Keep covered in the refrigerator after.
To serve, remove the cabbage from the liquid and enjoy it as a side to any dish. You can top it with some extra dill or finely chopped green onion.