Begin by buttering a 9x13-inch pan thoroughly to ensure the caramel doesn't stick. Set the pan aside for later.
In a large saucepan, melt the butter over medium heat. Once melted, add the sweetened condensed milk, corn syrup, and brown sugar to the pan. Stir the mixture constantly with a rubber spatula, making sure to scrape the sides and bottom of the pan regularly.
Continue stirring the mixture over medium heat until it starts to simmer. Maintain a constant simmer and keep stirring to prevent the caramel from burning. It's important to be patient and vigilant during this step.
Using a candy thermometer, monitor the temperature of the caramel. The ideal temperature range for reaching the softball stage is between 235°F and 245°F. If you don't have a candy thermometer, you can perform a simple water test. Fill a glass or bowl with cold water and add a drop of caramel from the pan. If the caramel forms a small round ball in the water and flattens when removed, it has reached the desired consistency.
Once the caramel reaches the softball stage, immediately remove the pan from the heat. Stir in the vanilla extract and salt until they are well incorporated into the caramel.
Pour the caramel into the prepared 9x13-inch pan and let it cool completely. It's important to exercise patience here, as the caramel needs time to set and firm up.
Once the caramel has cooled, you can cut it into squares. For individual servings, you can wrap the caramels in wax paper. This step adds a charming touch and makes them easier to enjoy on the go.