Beat butter, sugar, baking powder, baking soda, and salt together in a large mixing bowl on low 30 seconds.
Increase to medium and beat for 5 more minutes.
Add egg, sour cream, and vanilla, and mix until combined.
Add cake flour, and stir in just until incorporated.
Form the dough into a disk and wrap tightly in plastic wrap. Chill for at least 1 hour or until firm.
Toward the end of the chilling time, preheat oven to 375 F. Line 2 baking sheets with parchment or silicone baking mats.
On a lightly powdered sugared surface, roll out dough to ¼ inch thickness.
Use a 2 ½ inch or 3 inch circular cookie cutter to cut out the cookies, cutting the cookies as close to each other as possible. Gather the dough scraps and repeat the process.
Bake about 10-15 minutes, or until the edges and bottoms are just beginning to turn golden. You don't want the cookies to bake to golden brown.
Cool to room tempearature.
To make the frosting, combine, butter, powdered sugar, salt, 1 tablespoon cream, and vanilla in a bowl.
Beat on low to until just combined.
Increase to medium and beat until smooth, about 3 minutes, adding in more cream as needed.
Spread frosting over cooled cookies.
Store leftover cookies in the fridge.
Notes
The butter should be in between straight-from-the-fridge and room temperature, about 60F.