In a 2- to 3-quart pan over medium heat, stir 1/2 cup butter and the brown sugar frequently until butter is melted and mixture is smooth, about 3 minutes.
Meanwhile, in a large bowl, mix oats, flour, baking powder, and salt. Add butter mixture; stir until well blended.
Add egg and vanilla to oat mixture and stir until well blended.
Drop dough in 1-teaspoon portions about 2 inches apart onto cooking parchment-lined or buttered and floured 12- by 15-inch baking sheets.
Bake cookies in a 350° regular or convection oven until browned, 8 to 10 minutes; if baking two sheets at once in one oven, switch their positions halfway through baking. Let cookies cool on sheets for 5 minutes, then use a wide spatula to transfer to racks to cool completely.
Spread the bottom of each of half the cookies with about 1 tablespoon Chocolate-Mint Ganache. Top each with a second cookie, bottom toward filling, and press gently to squeeze filling to edges of sandwich.
Chocolate-Mint Ganache. In a heatproof bowl set over (but not touching) barely simmering water in a pan, occasionally stir 3 cups chopped semisweet chocolate (18 oz.) and 1/4 cup whipping cream until chocolate is melted and mixture is smooth. Remove from heat and stir in 3/4 teaspoon peppermint extract. Let cool until thick but not firm, about 15 minutes, and use immediately (see note).