Preheat oven to 350 degrees. Line 2 baking sheets with parchment. Sift flour, baking powder, baking soda, and salt into a bowl. Combine banana and sour cream in another bowl.
Beat butter and granulated and brown sugars with a mixer on medium-high speed, until pale and creamy, about 3 minutes. Add egg and 1/2 teaspoon vanilla, beating until incorporated. Add banana mixture in 2 additions, alternating with flour mixture.
Transfer batter to a pastry bag fitted with a 1/4-inch plain tip. Pipe batter into 1 1/4-inch rounds on baking sheets, spacing rounds 1 1/2 inches apart.
Bake until edges are golden, about 12 minutes. Slide parchment, with cookies, onto wire racks. Let cool. (Unfilled cookies can be stored for up to 1 day.)
Beat cream cheese, confectioners' sugar, and remaining 1/2 teaspoon vanilla with a mixer on medium speed until smooth, about 2 minutes.
Pipe or spoon 1 tablespoon cream cheese mixture onto the flat sides of half the cookies. Sandwich with remaining cookies. Dust with confectioners' sugar, and serve immediately.