In a medium bowl, sift the flour, baking powder, and salt. Whisk to evenly combine.
In a large bowl, using a handheld mixer, or in the bowl of a stand mixer, using the paddle attachment, cream the butter and sugar until light and fluffy, stopping the mixer once or twice to scrape down the bowl with a rubber spatula. Add the eggs, 1 at a time, beating after each until thoroughly incorporated. Add the vanilla. Scrape the bowl again and mix for a few more seconds. On low speed, add 1/2 the dry ingredients. Beat until combined. Add the remaining dry ingredients and beat just until the flour is completely absorbed into the dough.
Turn the dough onto the table, gather it together, and knead gently into a smooth mass. Form it into a log and divide into 3 even parts. Flatten each section into a patty about 1/2-inch thick and cover each with plastic wrap. Chill until firm, at least 2 hours or up to 3 days. The dough may also be frozen for up to 1 month.
Preheat oven to 350 degrees. Let the dough sit at room temperature, still covered, for about 20 minutes to soften slightly. On a lightly floured surface, roll the dough to slightly more than 1/8-inch thick.
Cut out shapes with cookie cutters, as close together as possible, and use a palate knife or metal spatula to transfer to ungreased baking sheets, placing them about 1-inch apart. Repeat with the remaining pieces of dough. Press the scraps together, roll them out, and cut, always brushing off any excess flour, until all used up.
Bake cookies for about 17 minutes, or until the edges begin to turn golden. Let sit for 5 minutes, then transfer to wire racks to cool completely.
In the bowl of a stand mixer fitted with the paddle attachment, or in a bowl using a handheld mixer, begin to beat the egg whites and sugar on low speed. Be sure the bowl and beaters are free of any fat and that there is no egg yolk in the whites. After 30 seconds or so, scrape down the bowl with a rubber spatula and gradually increase the speed of the mixer to medium. Beat until the mixer thickens and stiffens, about 3 minutes.
STORING ICING: Keep Royal Icing in an airtight container. To prevent a crust from forming while using the icing, keep the container covered with a damp towel or plastic wrap. Royal Icing will keep refrigerated for up to 2 days. Before reusing, lightly beat with a whisk or fork.
Yield: 2 1/2 cups icing to cover about 4 to 6 dozen 3-inch cookies
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.