Prepare Dough: In a bowl, mix together the flour, baking powder and baking soda; set aside.
Using an electric mixer, beat the butter and peanut butter. Add sugars and beat at medium speed for 3-4 minutes, scraping down bowl twice.
At low speed, add egg and vanilla. Add flour mixture and stir until well mixed, frequently scraping down bowl. Add peanuts (if using) and oats, and mix well. Using an ice cream scoop, scoop out 12 rounds of dough and put on plastic wrap lined dinner plates. Cover and chill for 30 minutes. After 30 minutes, shape dough into 3/4 inch thick rounds.
Preheat oven to 350 degrees F.
Arrange rounds on parchment lined baking sheets spacing a good 4 inches apart.
Bake until cookies have spread and turned very light golden brown, about 10-14 minutes. Remove from oven and set aside to cool and firm up, 5 to 10 minutes. Transfer to a rack to cool completely before filling.
Filling: Using an electric mixer, beat the butter, peanut butter and confectioners' sugar until very smooth.
Assembly: Spread a thin layer (about 1/8 inch) on underside of a cookie. Sandwich with another cookie. Repeat.