Lightly butter an 8 by 8 by 2-inch pan and line bottom with parchment paper or plastic wrap; set aside. In a medium heavy-bottomed saucepan, combine corn syrup, both sugars, milk, salt, and baking soda. Bring to a boil over medium-high heat, stirring occasionally. Insert a candy thermometer. Cook, stirring frequently, until temperature registers 236 degrees F (just below soft-ball stage); don't overcook.
Immediately remove saucepan from heat, stir in the chilled butter, then the peanut butter and vanilla until smooth. Pour mixture into prepared pan. Set aside in a cool, dry place, and let fudge set.
Before the fudge cools completely, cut into desired shapes. Set shapes aside to cool completely. Store fudge in an airtight container up to 1 week.
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