In a medium bowl, mix the flour, salt and baking powder. In a small bowl, lightly beat the egg with the extracts.
In the bowl of a standing mixer (or in a large bowl with a handheld mixer), beat the butter and sugar until fluffy, about 1 minute. With the mixer running, pour in the egg mixture and continue beating until incorporated. Scrape down the sides of the mixer bowl. With the mixer on low, gradually add the flour and mix just until evenly blended.
Form the dough into a ball, press it into a disc, and wrap tightly in wax paper. Refrigerate for 2 hours or overnight.
Line a baking sheet with parchment paper or a silicone baking mat. Adjust a rack to the middle position and heat the oven to 375F. Take the chilled dough out of the refrigerator and let it sit at room temperature for 5-10 minutes (longer if it was chilled overnight). On a lightly floured counter or sheet of wax paper, roll the dough out to ¼-inch thick. Use as little flour as possible on board and rolling pin to prevent sticking. Cut cookies using a floured cookie cutter. Re-roll scraps, always using as little flour as necessary.
Bake the cookies for 5-9 minutes, until they no longer look wet on top and the edges have just started to brown. The baking time will depend on the size of the cookies you’ve cut. Let the cookies rest for a on the baking sheets for 1-2 minutes before transferring them to cooling racks.
While cookies are cooling, make the royal icing.
Whisk together the water and meringue powder. Add the cream of tartar and whisk until completely combined.
Pour the mixture into the bowl of your stand mixer. Add the powdered sugar, and using the mixer paddle attachment, mix at a low speed for 10 minutes.
Tint with your desired color. If using multiple colors, divide frosting into separate bowls before adding food coloring. Keep bowls covered with a damp paper towel when not in use to prevent the frosting drying out.
Decorate cooled cookies with icing and whatever else (sprinkles, nuts, colored sugar, etc) you want.
Meringue powder is available at Micheals or a baking supply store. Vanilla extract is the most common flavoring for sugar cookies, but these are a good template for playing with extract flavors. Try using 1/2 teaspoon of two different extracts, like vanilla and lemon, or maple flavor in the cookie dough and root beer flavor in the icing, or whatever combination you like! These cookies are great for any occasion, just use whatever cookie cutter is appropriate.