For the Cookies
Preheat your oven to 400 degrees F. Line two baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment cream the butter and sugar until light and fluffy, about 2 minutes.
Add the eggs, vanilla and lemon extract (if using). Mix until incorporated.
With the mixer on low speed, add the flour, baking powder and salt. Mix until just incorporated.
Cover and refrigerate for at least one hour.
Roll the dough out to about 1/4 inch thick. Use cookie cutters to cut shapes and place the cookie dough 1 inch apart on the baking sheet.
Freeze the cut dough for 10 minutes and then bake 6 to 8 minutes.
Cool completely before frosting.
For the Frosting
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed for about 2 minutes, until light and fluffy.
With the mixer on low speed, gradually add the powdered sugar, heavy cream, vanilla and salt.
Turn the mixer to medium-high speed and beat the frosting for an additional 5 minutes, until nearly white.
Before frosting or piping, mix the frosting by hand with a wooden spoon or spatula to push out the air pockets.