Melt butter and stir in oats: In a saucepan over medium heat, melt ½ stick butter. Add oats and cook, stirring, until toasted, 5 to 10 minutes.

Spread and cool mixture: Spread mixture onto a parchment-lined baking sheet; let cool completely.

Combine flour, baking soda, and salt: In a bowl, whisk together flour, baking soda, and salt.

Beat remaining butter with sugars: Beat remaining 1 stick butter with granulated and brown sugars on medium-high speed until pale and fluffy, about 3 minutes.

Mix in peanut butter: Add peanut butter; beat until well combined.

Mix in oat mixture and flour mixture: Gradually add oat mixture and flour mixture; beat on low speed until combined.

Roll dough and refrigerate: Roll out dough between 2 sheets of parchment ¼ inch thick. Slide dough with parchment onto baking sheet and refrigerate until chilled, about 20 minutes.

Preheat oven and cut out cookies: Preheat oven to 350°F. Remove top layer of parchment; cut out cookies using a 2-inch round cookie cutter.

Transfer cookies to baking sheets: Place cookies on parchment-lined baking sheets, spaced about 1 inch apart; sprinkle with turbinado sugar.

Bake and cool: Bake until golden, about 10 minutes. Let cool completely on sheets.

Make filling: In a medium bowl, beat all ingredients on medium speed until smooth. Transfer mixture to a pastry bag fitted with a ½-inch round tip (such as Ateco #806).

Pipe filling onto cookies: Pipe filling in a spiral motion on bottom sides of half of cookies.

Sandwich cookies: Top with remaining cookies to form sandwiches.