In a medium saucepan over medium-low heat, sprinkle brown sugar evenly over the bottom of the pan.
As the sugar melts, stir it with a wooden spoon or whisk until it dissolves completely. Be sure to watch the sugar closely, as it can quickly turn from melted to burnt.
Once the sugar has melted and turned a dark amber color, remove the saucepan from the heat and add the unsalted butter.
Stir the butter into the melted sugar until it's completely melted and combined.
Add the milk slowly and continue to stir the mixture until it's combined.
Return the saucepan to medium heat and bring the mixture to a boil. Be sure to stir constantly with a wooden spoon to prevent the caramel from sticking to the sides of the pan.
When the caramel reaches 245°F, remove it from the heat. If you don't have a candy thermometer, you can test the caramel's thickness by spooning a small amount onto a cold plate. If the caramel cools to a thicker consistency, it's ready.
Once you've removed the caramel from the heat, add the vanilla extract and a pinch of salt. Stir the mixture well until the vanilla and salt are fully incorporated.
Once your caramel sauce has cooled to room temperature, transfer it to an airtight glass container, like a mason jar. Store the jar in the refrigerator for up to two weeks.