In large bowl, cream together butter, brown sugar, and white sugar until light and fluffy. Beat in vanilla extract and peanut butter. Beat in egg until well blended.
1 stick (½ cup) unsalted butter, ¾ cup light brown sugar, ¼ cup granulated sugar, 1 tablespoon vanilla extract, ½ cup creamy peanut butter, 1 large egg
Combine oats, flour, baking powder, baking soda, cinnamon, and salt. Add into peanut butter mixture and beat until just combined.
¾ cup quick oats, 1 cup all-purpose flour, ½ teaspoon baking powder, ½ teaspoon baking soda, ½ teaspoon cinnamon, ½ teaspoon salt
Chill cookie dough, covered, at least 2 hours and up to 1 week.
Preheat oven to 350 degrees F. Form rounded tablespoons of dough into balls and arrange about 2 inches apart on baking sheets lined with parchment paper.
Bake cookies in batches in middle of oven for 10 to 11 minutes, or until just pale golden. Cool cookies on baking sheet 5 minutes and transfer to racks to cool completely.
While cookies are cooling, prepare marshmallow buttercream filling.