Line an 8x8-inch pan with parchment paper, leaving overhang on two sides for easy removal.
Combine the peanut butter and butter in a large microwave-safe bowl. Microwave in 30-second intervals, stirring between each, until melted and smooth (about 1.5 minutes total).
Stir in the vanilla extract.
Gradually stir in the powdered sugar, one cup at a time, until the mixture is thick and smooth.
Pour into the prepared pan and spread into an even layer. Smooth the top with a spatula.
Refrigerate for at least 1 hour until firm, then cut into squares.
Notes
For the smoothest fudge, sift the powdered sugar before adding. Press firmly into the pan and smooth the top with the back of a buttered spoon.