Special equipment: cookie cutters
For the cookies: In the bowl of a stand mixer fitted with the paddle attachment, beat the butter until soft, then add the granulated sugar and beat until light and fluffy. Add the eggs, one at a time, beating after each addition and scraping down the sides of the bowl as needed. Add the vanilla and almond extracts.
In a medium bowl, whisk the flour, almond meal, baking powder, nutmeg and salt. Add the flour mixture to the mixer set on low speed, and mix until combined. Divide the dough into two discs, wrap tightly in plastic wrap and chill for at least 1 hour or up to overnight.
Line two baking sheets with parchment paper. Position racks in the middle and lower third of the oven. Preheat the oven to 350 degrees F. Lightly dust a clean surface with flour.
Working with one disk at a time, roll out the dough to 1/4-inch thick. Use cookie cutters to cut out desired shapes, and place the cookies 1-inch apart on the prepared baking sheets. Re-roll the dough scraps to make more cookies, about 48 total.
Bake the cookies until firm to the touch but not browned around the edges, 12 to 13 minutes, switching the pans from top to bottom and rotating them front to back halfway through cooking. Cool the cookies for 2 minutes on the baking sheets, then transfer to a wire rack to cool completely.
For the icing: In the bowl of a stand mixer fitted with the whisk attachment, whisk the egg whites and vanilla until frothy. Add the confectioners' sugar 1/4 cup at a time until smooth. Add the lemon juice and mix until combined.
Decorate the cookies with the royal icing. Store in an airtight container until ready to serve.
Use the royal icing immediately or store in an airtight container at room temperature for 3 to 5 days.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.