In a medium bowl, whisk together the flour, salt, and baking powder. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter, shortening, and sugar together until light and fluffy.
Add the egg and vanilla and beat until just combined. Add the flour mixture and mix until incorporated. Wrap the dough in plastic and refrigerate for at least 4 hours.
Preheat oven to 325 degrees F. Line two baking sheets with parchment paper.
Dust a clean work surface with flour. Roll the chilled dough until you reach 1/4 inch in thickness.
Use your favorite cookie cutter to cut shapes in the dough and transfer the cookies to the prepared baking sheets.
Bake for 12 minutes, until they are set but not browned. Remove from the oven and place the baking sheet on a cooling rack for 5 minutes. Transfer the cookies to the rack to cool completely.
Making the Basic Royal Icing:
In a large bowl, whisk together the sugar, egg whites, and lemon juice until the mixture is completely smooth. It should have the texture of a glaze (if it's too thin, add a bit more sugar. If it's too thick, add a few drops of lemon juice.)
Glaze cookies with a pastry bag fitted with a small medium tip or just by using a spoon. Let the icing harden before serving.
The cookies can be kept in an airtight container for up to 3 days.