Place room temperature butter and granulated sugar in the bowl of your mixer with the paddle attachment and mix on low until smooth. The creamed butter should be fluffy and pale yellow in color. Make sure to scrape the bottom of the bowl.
Add the egg and mix on medium (4 on a Kitchenaid mixer) until fully incorporated. Scrape the bowl as needed.
Add the vanilla extract and mix until just incorporated.
In a separate large bowl, whisk the flour, salt, baking powder, and nutmeg together.
Add the flour mixture into the dough in batches (about ⅓ of the total dry ingredients at a time) and mix until fully incorporated. Start on low until the flour begins to incorporate, then turn up to medium. Scrape the bowl as needed.
Add the milk once the dry ingredients are fully incorporated. Continue mixing on low until the dough becomes a solid mass.
Scrape out the mixer bowl and wrap the dough in plastic wrap.
Roll the wrapped dough into a flat square with a rolling pin, this helps it chill faster and more evenly.
Refrigerate the dough for about 20 minutes, or until firm to the touch.
Once chilled, knead the dough slightly and roll it out on a lightly floured surface to about ¼ inch thick.
Use a heart cookie cutter to cut out uniform hearts. Keep the rolled dough chilled if you're not cutting and baking right away, it gets harder to work with the warmer it gets.
Place the cookies on a parchment-lined baking sheet. Put the sheet back in the refrigerator for 10 to 15 minutes to chill again, this helps them avoid spreading.
Bake at 350º F for 10 to 14 minutes, depending on the size of your cookie. The cookies should be slightly golden brown at the edges but look done in the center.
Transfer the baked cookies to a cooling rack to cool completely before dipping them in the icing.
Making the Marbled Icing
Combine the egg whites, powdered sugar, and cream of tartar in the bowl of your stand mixer with the whisk attachment.
Mix on low until combined, then increase to high for 1 to 2 minutes. Add the vanilla extract and whip until bright white. No need to mix longer than 5 minutes.
Thin the icing by adding water and mixing on low speed until you reach your desired consistency.
Add water 1 teaspoon at a time until a ribbon of icing goes flat in about 15 seconds. This is a good dipping consistency, thin enough to flow but not so thin it runs off the cookies.
In a small bowl, color about ⅓ of your icing with your chosen color, or split it into multiple colors.
Drizzle the tinted icing over the top of the white icing.
Use a toothpick or skewer to drag up and down through the icing, creating a marbled effect.
Dip each cooled cookie into the icing, covering the edges evenly. Pull straight up and tap off any excess, letting it drip for a second, then twist to catch the tail and keep the edge clean.
Wipe the cookie edges clean if any icing dripped over the side. Place the frosted cookie on a wire rack or parchment-lined tray to dry, watching for drips.
Pop any bubbles in the icing with a straight pin or scribe, then give the cookie a gentle shake to even out the surface.
Let the cookies dry completely before packaging, this can take 16 to 24 hours depending on your environment. The top dries quickly, but underneath it can still be runny and will crack if touched too soon. For faster surface drying, place the tray in a warm (turned off) oven with the door open, the residual warmth helps the tops set faster without smudging.
Once fully dry, store the cookies in an airtight container at room temperature. Wrap any leftover dough well in plastic and freeze until ready to use.
Notes
Ingredient notes: Weigh your ingredients to avoid cookie failures, a digital kitchen scale makes this quick and gives consistent results every time. Practice mise en place (everything in its place), measure out your ingredients ahead of time and have them ready before you start mixing to reduce the chances of accidentally leaving something out.Pan options: No heart-shaped cutter on hand? Any cutter shape works with this dough and icing technique, just adjust your bake time slightly for larger or smaller cutouts.Make-ahead and storage: Dough freezes well, wrapped tightly, for up to 3 months. Undipped baked cookies keep at room temperature in an airtight container for about a week. Leftover royal icing keeps refrigerated in an airtight container for up to a week, remix and adjust the consistency before reusing it.Substitutions: Any extract flavor works in the icing in place of vanilla. Use whatever gel food coloring brand and color combination you like for the marbling, the technique works the same no matter the color.Critical do-nots: Don't skip chilling the dough before rolling, it will be too soft to cut clean shapes and the cookies will spread and warp in the oven. Don't rush the icing consistency, too thin and your colors will bleed together instead of staying marbled. Don't touch or stack the cookies until the icing is fully dry, or you risk cracking the shell.