Preheat the oven to 350°F (175°C). Lightly grease a 10x15-inch rimmed baking sheet or line with parchment paper for easy removal.
In a large mixing bowl, blend together the butter, peanut butter, granulated sugar, and brown sugar until the mixture becomes light and fluffy. Add the eggs individually, beating well after each addition. Stir in the vanilla extract.
In a separate bowl, whisk the flour with baking soda and salt. Slowly incorporate the dry mixture into the creamed mixture, stirring until just combined. Fold in the oats to complete the batter.
Evenly distribute the batter across the prepared pan, smoothing the surface. Bake in the preheated oven for 20 to 22 minutes, or until the edges are lightly browned and a toothpick inserted into the center emerges clean. Avoid overbaking.
Remove from the oven and allow the bars to cool entirely in the pan before frosting.
For the frosting, cream together the peanut butter and butter in a bowl until smooth. Add the vanilla, then gradually beat in the powdered sugar, alternating with milk until the mixture becomes fluffy and spreadable.
Once the bars have cooled, evenly spread the frosting across the surface. Slice into squares or rectangles and serve.