Mix together the goat’s milk, sugar, corn syrup, and cinnamon stick in a large pot (at least 3 quarts). Bring to a simmer over medium-high heat, stirring so the sugar dissolves.
Remove the pot from the heat and add the dissolved baking soda, stirring all the while. The mixture should froth, but if it doesn’t your goat’s milk just might not be very acidic (that’s okay). Stir it until most of the foam subsides.
Return the pot to medium-high heat and bring to a brisk simmer. Cook for an hour or so, or until the mixture turns golden. Stir every five minutes or so to make sure it isn’t scorching on the bottom of the pan.
After the mixture turns pale golden, you’ll need to start stirring more frequently. The mixture will darken to a caramely-gold color and thicken to the consistency of maple syrup--it should easily coat the back of a spatula or spoon. This should take another 15 minutes or so, but you can continue to cook until it is the color and thickness of your liking.
Strain the hot caramel mixture through a sieve into a glass jar. Remove the cinnamon stick from the sieve (If you don’t have a sieve, it’s okay--you nay find a lump or two in the finished caramel).
If you want to finish with liquor, pour into the warm caramel and stir.
The cajeta will thicken as it cools, and will keep in the refrigerator in a covered container for at least one month. If it is thicker than desired when cool, you can add warm or hot water to thin.