Beat butter and eggs until blended. Beat in cake mix and oats. Refrigerate, covered, 2 hours or until firm enough to roll, though dough will remain fairly soft.
Preheat oven to 350°. On a well-floured surface, roll half of dough to 1/4-in. thickness. Cut with a floured 2-1/2-in. round cookie cutter. Place 1 in. apart on parchment-lined baking sheets. Bake until set, 8-10 minutes. Remove from pans to wire racks to cool completely. Repeat with remaining dough.
Spread frosting on bottoms of half of the cookies; cover with remaining cookies.
Freeze option: Freeze assembled sandwich cookies in freezer containers, separating layers with waxed paper. To use, thaw before serving.
Test Kitchen Tips
Try rolling out your dough between two sheets of lightly floured waxed paper.
If you have a go-to vanilla buttercream recipe, feel free to swap it for the pre-made.