Preheat oven to 350°F (180°C). Line 3 rimmed baking sheets with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugars at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in egg. Add banana and vanilla bean paste; beat until combined.

In a medium bowl, whisk together flour, cornstarch, salt, baking soda, cinnamon, and ginger. Add flour mixture to butter mixture all at once; beat at medium-low speed until nearly combined, stopping to scrape sides of bowl. Add oats, and beat just until combined. Scoop dough by 2 tablespoonfuls (about 36 grams each), and roll into balls. Place at least 1½ inches apart on prepared pans. (Your hands will get a bit messy, but it will be worth it.)

Bake, one batch at a time, until 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.

Place Blueberry Cream Cheese Frosting in a large pastry bag fitted with a medium French star piping tip (Wilton No. 199). Pipe frosting onto flat side of half of cookies. Place remaining cookies, flat side down, on top of frosting. Serve immediately.