Start by greasing the square baking pan and lining it with parchment paper, allowing the paper to hang over the edges for easy removal later.

Combine the sweetened condensed milk, unsalted butter, sugar, and a pinch of salt in the heavy-bottomed saucepan. Cook over medium heat, stirring constantly until the mixture comes to a boil.

Insert the candy thermometer into the mixture and continue to cook, stirring frequently. Let it simmer until it reaches the softball stage (235°F to 240°F or 118°C to 120°C).

Once the mixture reaches the softball stage, remove it from the heat and add the vanilla extract. Beat the fudge vigorously with the wooden spoon until it thickens and loses its gloss.

Pour the thickened fudge into the prepared baking pan. Smooth the top with the back of the spoon. Let it cool at room temperature until firm, then refrigerate for a few hours before cutting into squares.