Combine sugar, milk, butter, and salt in a small Dutch oven; cook over medium heat, stirring constantly, until sugar dissolves. Cover and continue to cook over medium heat 2 to 3 minutes to wash down sugar crystals from sides of pan. Uncover and cook, without stirring, until mixture reaches soft ball stage (236°).
Remove from heat, and stir in peanut butter and vanilla. Beat at medium speed of an electric mixer until fudge begins to harden; pour into a buttered 8-inch square baking pan. Mark top of warm candy into 1-inch squares, using a sharp knife. Cool completely before cutting.