Preheat oven to 350°F. Lightly grease 2 large cookie sheets.
In a small bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, beat together the butter and peanut butter with an electric mixer on medium speed until smooth. Beat in sugars until well blended. Beat in egg and vanilla. With mixer on low speed, beat in flour mixture until just blended. Stir in oats.
Scoop dough onto prepared cookie sheets, 1 level tablespoonful at a time, about 2 inches apart. Press cookies flat with the back of a floured fork, to about ¼-inch thickness. Bake in preheated oven until golden brown at edges, about 12 minutes. Remove from oven and let cool on cookie sheet 1 minute before transferring to a wire rack to cool completely.
Meanwhile, in a bowl, with mixer on medium speed, beat sugar, cream cheese, and peanut butter until smooth. Spread 1 tablespoon of the peanut butter mixture onto flat side of 16 cookies, and top each with one of the remaining cookies, flat side down, to make a sandwich.
244 calories (7g protein, 27g carbohydrate, 12g fat) per serving