Beat butter and sugars with a mixer at medium speed until well blended. Add vanilla and egg; beat well. Add peanut butter; beat until blended. Divide butter mixture evenly between 2 bowls.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking soda, and salt. Spoon half of flour mixture into 1 bowl of butter mixture; stir until well blended.
Stir cocoa into remaining flour mixture; add to remaining bowl of butter mixture, stirring until well blended. Cover both bowls of dough, and chill at least 1 hour.
Preheat oven to 375°.
Roll peanut butter dough into 76 (1/2-inch) balls. Roll chocolate dough into 76 (1/2-inch) balls. Gently roll together 1 chocolate dough ball and 1 peanut butter dough ball. Place balls 2 inches apart on baking sheets coated with cooking spray. Flatten cookies with the bottom of a glass. Bake at 375° for 8 minutes. Cool 1 to 2 minutes on pans. Remove from pans; cool completely on wire racks.