Peanut Butter Sandwich Cookies

1/2 cup plus 4 tablespoons unsalted butter, room temperature, divided (1 1/2 sticks)
1 cup old-fashioned rolled oats
1 cup plus 2 tablespoons all-purpose flour
1 teaspoon baking soda
1 teaspoon salt
1/2 cup packed dark-brown sugar
1/3 cup granulated sugar
1/2 cup smooth peanut butter
Turbinado sugar, for sprinkling
For the Peanut Butter Filling:
3/4 cup peanut butter
4 tablespoons unsalted butter
1/4 cup confectioners' sugar
1/2 teaspoon salt

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 10 minutes. Spread oat mixture onto prepared baking sheet and let cool.
Meanwhile, preheat oven to 350°. Line 2 baking sheets with parchment paper or a nonstick baking mat; set aside.
In a medium bowl, whisk together flour, baking soda, and salt; set aside. In bowl of an electric mixer fitted with the paddle attachment, cream together remaining 1/2 cup butter, brown sugar, and granulated sugar. Add peanut butter and mix until well combined.
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 baking sheet and transfer to refrigerator until chilled, about 20 minutes.
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.
Transfer baking sheets to oven and bake until golden, rotating baking sheets halfway through baking, about 10 minutes. Let cool completely on baking sheets.
When cookies have cooled completely, snip end of a pastry bag and squeeze peanut butter filling in a spiral motion on bottom side of half the cookies. Top with remaining cookies to form sandwiches.

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