Prepare the Pan: Line an 8x8 inch baking pan with parchment paper, leaving a slight overhang on each side for easy removal. Lightly grease the parchment with butter to prevent sticking.
Melt the Butter: In a large, heavy-bottomed saucepan, melt the butter over medium heat without browning it.
Combine Ingredients: Add the brown sugar, corn syrup, sweetened condensed milk, and salt to the melted butter. Stir until the ingredients are well combined.
Bring to a Boil: Increase the heat slightly to bring the mixture to a gentle boil, then reduce to medium-low. Attach a candy thermometer to the pan and cook, stirring frequently, until the caramel reaches 245°F (118°C), the firm ball stage.
Add Vanilla: Remove the pan from heat and stir in the vanilla extract.
Pour and Set: Carefully pour the hot caramel mixture into the prepared pan, using a spatula to spread it evenly. Allow it to cool at room temperature for at least 2 hours, or until fully set.
Cut into Squares: Once set, lift the caramel out of the pan using the parchment paper overhang, then use a sharp knife to cut it into small squares.
Optional Sea Salt Topping: For a finishing touch, sprinkle the caramels with sea salt just before they set fully. The salt enhances the caramel's sweet flavor.