Preheat the oven to 350 degrees F (175 degrees C).

Beat 1/2 cup butter, 1/2 cup peanut butter, white sugar, brown sugar, and vanilla in a large bowl until light and creamy. Add egg and beat until fully incorporated.

Whisk flour, baking soda, baking powder, and salt together in a separate bowl. Add flour mixture to the peanut butter mixture and stir until just combined. Stir in oats until evenly incorporated.

Drop dough by teaspoonfuls onto ungreased baking sheets. Flatten each mound with a fork to form 1/4-inch-thick cookies.

Bake in the preheated oven until edges are lightly browned and centers are set, about 10 minutes. Let cool on the baking sheets briefly before transferring to wire racks to cool completely.

To make the filling: Cream 3 tablespoons butter with powdered sugar, 1/2 cup smooth peanut butter, and heavy whipping cream. Spread filling onto half of the cooled cookies, then top with the remaining cookies to form sandwiches.