Preheat oven to 350°F (180°C). Line 2 baking sheets with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, ¾ cup (150 grams) granulated sugar, and brown sugar at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in egg and vanilla bean paste.

In a medium bowl, whisk together flour, ½ cup (36 grams) ground cereal, salt, cardamom, baking soda, and cream of tartar. Add flour mixture to butter mixture all at once; beat at medium-low speed just until combined.

In a small bowl, stir together remaining 2 tablespoons (24 grams) granulated sugar and remaining 1 tablespoon (4.5 grams) ground cereal.

Scoop dough by 1½ tablespoonfuls (about 26 grams each), and roll into smooth balls; toss in sugar mixture. (Add more ground cereal to sugar mixture for final few dough balls, if desired.) Place at least 2 inches apart on prepared pans.

Bake, one batch at a time, until lightly golden, edges and tops are just set, and centers still seem slightly underdone, 8 to 10 minutes, rotating pan halfway through baking. Let cool on pan for 5 minutes. Remove from pan, and let cool completely on wire racks.

Spoon White Chocolate Buttercream into a pastry bag fitted with a medium open star piping tip (Wilton 1M). Pipe buttercream onto flat side of half of cookies. Place remaining cookies, flat side down, on top of buttercream.