1. In a large bowl, mix flour, baking soda and salt.
2. Beat butter and peanut butter until smooth. Gradually add sugar, beating to blend. Beat in egg and vanilla. Stir in flour mixture and butterscotch morsels. Form into a large disk and wrap in plastic. Chill for 2 hours or overnight.
3. Preheat oven to 375°F; line 2 baking sheets with parchment. Use hands to form dough into 1 1/2-inch balls. Place balls about 2 inches apart on cookie sheets. Using a fork, press balls flat, then press again to form a crisscross pattern. Bake cookies until lightly browned, about 15 minutes. Cool on sheets for 10 minutes, then transfer to racks to cool completely. Repeat with remaining dough.