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Line an 8x8-inch pan with parchment paper or an aluminum foil sling for easy removal. If using aluminum foil, be sure to butter it so the fudge will come off easily.
Combine brown sugar, milk, and butter in a medium saucepan over medium-high heat. Cook while stirring frequently until the brown sugar and butter dissolve and the mixture comes to a boil.
Boil for 2 minutes, then remove from heat.
Stir the peanut butter and vanilla extract into the mixture until smooth and well combined.
Add the powdered sugar to a large bowl, then pour the hot peanut butter mixture over it.
Beat together with a hand mixer until combined and there are no lumps of powdered sugar remaining.
Transfer the fudge to the prepared pan, and let it set at room temperature for about 3 hours.
Allow it to cool completely before slicing it into squares.