Line an 8x8-inch pan with parchment or foil, leaving an overhang, and butter it lightly. Have the peanut butter, vanilla, butter, and salt measured and waiting by the stove.

Combine the sugar and milk in a heavy medium saucepan. Stir over medium heat until the sugar dissolves and the mixture comes to a boil.

Stop stirring and boil until the mixture reaches 234°F on a candy thermometer (softball stage), about 4–5 minutes of boiling. If you have no thermometer, drop a little syrup into cold water — it should form a soft ball that flattens when you pick it up.

Remove the pan from the heat at once. Add the peanut butter, butter, vanilla, and salt, and beat vigorously with a wooden spoon until the fudge thickens, turns matte instead of shiny, and starts to hold its shape — 2 to 3 minutes. Work quickly once it starts to set.

Scrape the fudge into the prepared pan and smooth the top with a buttered spatula.

Let it set at room temperature for about 30 minutes, then lift it out and cut into 36 small squares.