For the cookies: Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper and set aside.
In a medium bowl, whisk together the flour, baking powder and salt. In a large bowl with an electric mixer or the bowl of a stand mixer fitted with a paddle, beat together the butter and granulated sugar on medium-high until pale and fluffy, 3 to 4 minutes. Reduce the speed to medium and add the extracts and egg and mix to combine. Reduce the speed to low and then gradually add the dry ingredients and mix to combine. Pour the dough out onto a clean work surface and give it a few kneads to bring it all together. Divide it in half, wrap half of it in plastic wrap and refrigerate it while you roll out the first half. (Alternatively, you can make the dough in advance, wrap all of it in plastic wrap and refrigerate it overnight.)
Dust your work surface and a rolling pin with flour and roll out the dough to 1/4-inch thickness. Cut out 5 increasing sizes of circles, 1 1/2 inches to 3 inches, and use a small offset spatula to transfer them to a baking sheet, 1-inch apart. Re-roll the scraps and repeat with the remaining half of the dough. Bake until the bottoms are lightly browned, 10 to 12 minutes. Let the cookies cool on the pans for 5 minutes and then transfer to a wire rack to cool completely.
For the frosting: In a large bowl with an electric mixer or the bowl of a stand mixer fitted with a paddle, beat the butter until soft and creamy. Gradually add the powdered sugar and mix to combine. Mix in the extracts and salt. Add food coloring if desired.
To assemble: Transfer the frosting to a piping bag. Stack up the circles, largest on the bottom, with a thin layer of frosting between them and decorate with sprinkles as desired.