Line a 9 X 13 inch baking dish with foil or parchment paper. Spray well with nonstick cooking spray.
Place the sweetened condensed milk, brown sugar, and salt in a large 6-8 quart sauce pot and set over medium-high heat. Secure a candy thermometer to the side of the pot. Stir well and bring to a boil.
Once boiling, lower the heat to medium-low. Stir continually until the candy thermometer reaches "soft ball" temperature, or 240 degrees F. It's okay if the bottom browns and little burnt sugar pieces are floating around the mixture. Just make sure to keep stirring!
At 240 degrees F, remove from heat and immediately stir in the butter and vanilla extract. Once smooth, carefully pour the mixture into the prepared dish and allow the mixture to cool.
When the Mexican Burnt Milk Candy is firm, but still warm, cut into small squares. Do not wait to cut the candy when it's all the way cool. It becomes hard to cut.