Preheat your oven to 350°F. In the bowl of a stand mixer, beat the butter, peanut butter, and sugars on medium-high speed until light and creamy, about 5 minutes. Add in the egg and vanilla and mix well. Next, add the baking soda and salt and mix again. Finally, add the flour and oats and mix until the dough comes together and no more visible patches of flour can be seen. The dough is going to be really sticky - don't add any more flour. 

Scrape the dough out into an ungreased quarter sheet pan. You could also use a 9x13 pan, your bars will just end up thicker. Press the dough evenly into the pan using a rubber scraper. It's super sticky, so just do your best to make it even. Bake for 12-13 minutes, or until the edges just start to barely brown.

Top the still hot bars with the chocolate chips. Let them sit for a few minutes, then spread the melted chips evenly over the bars. Let the chocolate cool completely. Feel free to speed this step up by putting the bars in the fridge once the cookie portion has cooled. 

To make the frosting, beat the butter with a stand mixer until light and "whipped" looking. Add the peanut butter, and whip for 1 more minute. Add the powdered sugar, vanilla, and salt and mix well. Add the milk a little at a time, beating after each addition, until the frosting is spreadable, smooth, and fluffy. Top the cooled bars with the frosting, cut into 1 x 3-inch pieces, and enjoy!