Beat the butter and granulated sugar in the bowl of a mixer with the paddle attachment on low until smooth. Scrape the bottom of the bowl.
Add the egg, vanilla extract, milk and pink gel food coloring and mix on on medium speed until well mixed. If you want a deeper pink, add more food coloring. Scrape the bowl bottom and sides as needed.
In a separate bowl, whisk the flour, salt, and baking powder. Add 1/3 of the flour mixture into the dough and mix until fully incorporated. Start the mixer on low, then increase the speed to medium. Scrape the bowl bottom and sides as needed.
Add another 1/3 of the flour mixture and mix it again until incorporated. Scrape the bowl bottom and sides as needed.Finally, add the rest of the remaining flour mixture and mix it again until incorporated. Scrape the bowl bottom and sides as needed.
Work the dough into a disk and wrap it in plastic wrap. Chill in the fridge for about 20 minutes.
Once the dough has chilled, knead it slightly and, using a rolling pin, roll it on a lightly floured surface until about ¼ inch thick.
Use heart cookie cutters to cut out heart shapes. Arrange the cookies on a baking sheet lined with parchment paper. Refrigerate the cookie sheet for 10-15 minutes.
Bake the cookies in the oven to 350 degrees F for 10-14 minutes, until slightly golden brown on the edge.
Transfer the baked cookies to a cooling rack for icing.
In the bowl of a stand mixer fitted with the whisk attachment, combine the egg whites, powdered sugar, and cream of tartar.
Mix on low for about 1 minute and then on high for 1-2 minutes. Add the vanilla extract and whip until the mixture is a glossy white. This will take about 5 minutes.
The icing will be thick, so add the water 1 teaspoon at a time, and mix on low speed until you get to the desired consistency.
Spread the royal icing onto the cooled heart cookies, place back on the cooling rack, and IMMEDIATELY top with sprinkles.