Sugar Cookies: 

1. Preheat the oven to 350°F (177°C). Line a sheet tray with parchment paper or a silicone baking mat.
2. In a stand mixer fitted with a paddle attachment (or a large bowl with a hand mixer), cream together the softened butter and granulated sugar until fluffy, about 3-5 minutes. Add in the beaten eggs, vanilla extract, and milk and mix well to combine, scraping the bowl well after each addition.
3. In a separate large bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Add the dry ingredients to the wet ingredients and mix until just combined, scraping as needed.
4. Tightly wrap the cookie dough in plastic wrap, then refrigerate for at least 30 minutes before rolling it out.
5. After the dough has chilled, lightly flour your countertop and roll the cookie dough out to ¼-inch thickness. Cut out the cookies using your desired cookie cutters, then transfer them to the prepared baking sheet.
6. Bake the cookies in the preheated oven for 8-12 minutes, or until they are lightly golden brown around the edges.
7. Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. Make sure the cookies are completely cool before decorating.

Royal Icing: 

1. In a stand mixer fitted with a whisk attachment (or a large bowl with a handheld mixer), whisk together the egg whites and vanilla extract until white and foamy, then gradually add in the powdered sugar until the mixture forms stiff peaks. Finally, add in water one teaspoon at a time until you can make a figure eight with the icing that disappears in eight seconds.
2. Keep a large portion of the icing white, then color a small portion of the icing red and another small portion blue. Transfer the red and blue icing to their own piping bags, then drizzle the red and blue icing in straight lines over the white icing. Run a toothpick perpendicular to the lines to make a marbled pattern.
3. Dip the cookies in the icing one at a time, letting some of the icing run off before flipping the cookie over and transferring it to a wire rack to dry. If desired, top the cookies with sprinkles before they dry. Add more red and blue icing into the bowl periodically to add more color back to the marbled icing design. 
4. Make sure to let the royal icing dry completely before serving, then store the cookies in an airtight container at room temperature for up to a week.