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Line a baking sheet with parchment paper; set aside. Melt 4 tablespoons butter in a medium saucepan over medium heat. Add oats and cook, stirring, until toasted, 5 to 7 minutes. Spread oat mixture onto prepared baking sheet and let cool.
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Meanwhile, preheat oven to 350 degrees. Line 2 baking sheets with parchment paper or a nonstick baking mat; set aside.
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In a medium bowl, whisk together flour, baking soda, and salt; set aside. In the bowl of an electric mixer fitted with the paddle attachment, cream together remaining 1/2 cup butter, granulated and brown sugars. Add peanut butter and mix until well combined.
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Add oat mixture and flour mixture; mix on low speed until combined. Turn dough out onto a piece of parchment paper. Cover dough with a second piece of parchment paper and roll out dough, between parchment, to 1/4 inch thick. Carefully slide rolled dough onto the back of a baking sheet and transfer to a refrigerator until chilled, about 20 minutes.
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Remove top layer of parchment paper; cut out cookies using a 2-inch round cookie cutter. Place cookies 1 inch apart on prepared baking sheets; sprinkle with turbinado sugar.
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Transfer baking sheets to oven and bake until golden, rotating baking sheets halfway through baking, about 10 minutes. Let cool completely on baking sheets.
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When cookies have cooled completely, snip the end of the pastry bag and squeeze filling in a spiral motion on the bottom side of half of the cookies. Top with remaining cookies to form sandwiches.