Mix peanut butter, sweetened condensed milk, and salt; microwave briefly and chill 15–20 minutes.
Preheat oven to 350°F (175°C) and line two baking sheets with parchment.
Cream butter and sugars, add egg and vanilla; fold in flour, baking soda, salt, butterscotch morsels, and white chocolate.
Scoop dough, place 1 tsp chilled fudge inside, seal, and form balls; place 2 inches apart.
Bake 10–12 minutes until edges are golden; cool 5 minutes before transferring to a wire rack.