Add the 1/2 gallon of milk to a large pot with a lid over medium-low heat. Bring the temperature of the milk to 180 degrees Fahrenheit, stirring every few minutes to keep the bottom from scalding. This process might take about 20-25 minutes. Once to temperature, remove from the heat.
Let the milk cool to 110 degrees Fahrenheit. This may take 30-45 minutes. Continue to stir it occasionally and check the temperature. To speed this up, position the pot into a cold water bath.
In a small separate bowl, add the 1/3 cup of yogurt and one cup of the warm milk. Stir to combine.
Pour the yogurt mixture into the rest of the warm milk and stir.
Cover the pot with a lid and place it into the oven with only the light turned on. Let the yogurt sit for 24 hours undisturbed.
Once the 24 hours have passed, remove the pot from the oven. At this point, you will have a whole pot of thick plain yogurt.
Place a fine mesh strainer over a large bowl (to catch the whey), then line the strainer with cheesecloth or a clean tea towel. Pour the yogurt into the lined strainer, cover with plastic wrap or another towel to keep out contaminants, and place into the fridge. Strain for at least 10 hours. The longer you let the yogurt strain, the thicker the Greek yogurt will become.
Once the desired consistency is reached, enjoy as desired!
Notes
Store yogurt in an airtight container in the refrigerator for up to two weeks.
You might keep a note on your oven as a reminder that yogurt is inside. It can be easy to forget the yogurt inside and start to preheat the oven for something else.
The 24 hours in the oven could be reduced, but the full time gives a stronger, tangy yogurt taste.
Don't forget to put aside a little bit of yogurt as the yogurt starter for your next batch! Your cultures may weaken after a few batches, so just start fresh with another small amount of store-bought yogurt.
I tend to keep my Greek yogurt plain, but if you'd like to add sweetener to yours, add it in when you are stirring in your tempered yogurt starter. You can use maple syrup, honey, condensed milk, or regular sugar so long as you stir it gently to dissolve it.