Whisk together the flour, baking powder, and salt in a medium bowl.
Place the butter in a mixing bowl and use an electric mixer fitted with a paddle attachment. Mix on medium speed until creamy, about 1 minute. With the mixer running, gradually add the sugar and mix until the butter is lighter in color, about 1 minute total.
Whisk together the egg and vanilla in a small bowl. Return the mixer to medium speed, add the egg mixture, and mix until incorporated, about 30 seconds. Stop the mixer and scrape down the sides of the bowl and the paddle. Turn the mixer to low speed, gradually add the flour mixture, and mix until just incorporated, about 1 minute total.
Scrape the dough out onto a piece of plastic wrap and refrigerate until firm enough to roll, about 2 hours.
Preheat the oven to 375 degrees F. Have ready large baking sheets.
Flour a work surface and a rolling pin. Unwrap the dough and set the plastic wrap aside. Place the dough disk on the work surface and lightly dust both sides with flour. Roll out the dough then use the desired cookie cutters to cut out the. Using a flat spatula, transfer the cookies to the prepared baking sheets, leaving at least a 1/2 inch of space between them.
Bake 10 minutes or until just golden.
Remove from the oven and let the cookies sit on the sheets for about 1 minute. Using the flat spatula, remove the cookies to a wire rack to cool completely.
Combine ingredients in the bowl of a stand mixer fitted with a whisk attachment. Whisk on medium speed, stopping to scrape down the bowl, until glossy and stiff, about 5 minutes. Add additional lemon juice until the frosting is the desired consistency. Use food coloring for desired color.
Fill a piping bag or a resealable plastic bag with one corner snipped off and use to decorate cookies.