Preheat oven to 350.
Melt butter over medium heat, add oats and cook 5-7 minutes. Stirring often until toasted. Pour oats onto cookie sheet and let cool.
Combine the remaining 1/2 cup butter and sugars in a stand mixer with the paddle attachment and mix until no chunks of butter remain.
Add peanut butter and mix until combined, add oats then flour, soda & salt and blend until smooth. Chill dough in freezer for 5 minutes.
Transfer the dough to a large surface covered with parchment paper. Add a light dusting of flour to keep from sticking. Top with another large sheet of parchment and roll dough until 1/4" thick.
Cut the dough into 2.5" cookies and space 1/2" apart on an ungreased cookie sheet.
Bake for 20 mins, turning the pan halfway through baking, then transfer to a cooling rack.
Combine butter, salt, sugar and peanut butter in the bowl of a stand mixer and mix on medium speed until filling is smooth.
Place half the cookies with the top side down. using a piping bag or spoon, distribute the filling and close with another top, pressing down.