Preheat your oven to 325°F. Line rimmed baking sheets with unbleached parchment paper. Set aside.
In a large bowl, place the flour, xanthan gum, processed oats, baking soda, salt, granulated sugar, and brown sugar, and whisk to combine well.
Chop the butter into large chunks, and work it into the dry ingredients by pressing the butter down with the bottom of the bowl of a mixing spoon. Break up any lumps in the brown sugar at the same time, and keep working until the mixture resembles coarse, moist crumbs.
Create a well in the center of the mixture, add the egg, and vanilla, and mix to combine. The dough should be thick and relatively stiff.
Scoop the dough into about 12 pieces of relatively equal size, about 2 ounces each. Roll each into a smooth ball about 1 1/2-inches in diameter.
Place the pieces of dough 2 inches apart on the prepared baking sheets. Bake in the center of the preheated oven until the cookies are lightly golden brown all over and set in the center (12 to 14 minutes).
Allow the cookies to cool on the baking sheets until stable (about 10 minutes) before transferring to a wire rack to cool completely.
Make the filling
In the bowl of a stand mixer fitted with the paddle attachment, or a large bowl with a hand mixer, place the butter, milk, and vanilla, and beat on medium speed until well-combined. Raise the speed to high and beat until creamy (about 2 minutes).
Add the salt and about 3 1/2 cups of the confectioners’ sugar. Mix slowly until the sugar is incorporated.
Beat at high speed until uniformly thick. It's done when it's soft enough to pipe easily but still stiff enough to hold its shape (about 2 minutes more).
Beat in the meringue powder and as much of the remaining 1/2 cup of the confectioners’ sugar as necessary to thicken the frosting. If it gets too thick, add add more milk by the teaspoonful, beating well after each addition,
Assemble the pies
Transfer the filling to a pastry bag fitted with a 1/2-inch plain piping tip.
Once the cookies are completely cool, place half of them on a flat surface, inverted.
Pipe about 3 tablespoons of filling on the bottoms of the inverted cookies in the pattern of a 3 leaf clover (or in one large dollop).
Top with the other cookies to sandwich, and serve.
Notes
Picking a gluten free flour blend I recommend Better Batter's original blend without adding xanthan gum, Nicole's Best multipurpose plus xanthan gum, Bob's Red Mill 1-to-1 gluten free (in the blue bag) with an added 1/8 teaspoon xanthan gum so the cookies don't crumble. For a DIY version of Better Batter or the original Cup4Cup blend, which would work well here, too, see our all purpose gluten free flour blends page.