In a heavy-bottomed saucepan, combine the granulated sugar, brown sugar, milk, and butter. Cook over medium heat, stirring constantly, until the sugars are dissolved and the mixture comes to a rolling boil.

Once boiling, continue to cook without stirring until the mixture reaches 234-238°F (112-114°C) on a candy thermometer, which is the soft-ball stage. Remove the saucepan from the heat immediately.

Stir in the creamy peanut butter and vanilla extract until completely smooth and well combined. Work quickly as the fudge will start to set.

Pour the hot fudge into a buttered 9x13 inch pan and spread evenly. Allow it to cool completely at room temperature, which may take several hours, or chill in the refrigerator to speed up setting.

Once the fudge is firm, cut it into small squares or desired shapes. If desired, lightly dust with powdered sugar before serving for an extra touch.