Pour the entire gallon of milk into a large pot. Set it to warm slowly over low-medium heat until the milk is very frothy and steaming. Check with a temperature probe, and heat to 190℉. Do not let it boil. This can take some time, up to 30 minutes. Stir occasionally to help keep the milk from cooking onto the bottom of your pot.
Once the milk reaches 190℉, remove from the heat and partially cover for 1 hour. Allow the milk to cool to approximately 100℉ (10 degree difference either way is fine). If you don't have a temperature probe, you should be able to put your finger in the milk for 10 seconds. If you can't, it's too hot. If a skin develops, just remove it and discard.
After the milk has cooled, add the ½ cup yogurt. Stir to distribute, and then cover. Set the whole pot aside for 12-18 hours.
After 12-18 hours, open the pot and ensure your yogurt has thickened appropriately. Then stir in your sweetener of choice and vanilla if you're using it.
Ladle into jars and store in your refrigerator for up to 2 weeks.