First, pour 3 1/4 cups of water into the empty slow cooker and use a ruler to measure the depth. Note this measurement, as this is the depth your milk will be when it is finished.
Pour water out and place the milk and the liquid sugar in the slow cooker and stir to combine. Heat on low for 19-21 hours with the lid off. (If you're using a crystalline sugar, whisk it regularly during the first hour after it begins to steam in order to fully dissolve the crystals. Remove the skin that forms on the top periodically, especially toward the end of cooking, but don't scrape up any milk that cooks onto the sides.)
When the sweetened condensed milk has reached the desired depth, pour into a large bowl and add the butter and the vanilla. Once the butter is melted, whisk well and pour into a clean jar or pitcher and use in your favorite recipes. Store in the refrigerator for up to one week.
Notes
Note: Using a liquid sugar is the easiest method by far when using a slow cooker to make homemade sweetened condensed milk, but liquid sugars tend to be highly flavorful as well, so they may not be appropriate for your final outcome. I find maple syrup provides my favorite, most multi-functional sweetened condensed milk, but if you need something more flavor-neutral, use an evaporated cane sugar.