In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugar together until light and fluffy. Beat in the egg and vanilla and mix well.
In a separate bowl, whisk together the flour, nutmeg and salt. Slowly beat the dry ingredients into the butter mixture until combined. Wrap the dough in plastic and refrigerate for at least 1 hour.
Preheat oven to 350 degrees. Line a baking sheet with parchment paper.
Roll the dough out onto a flour surface and cut into desired shapes. Alternatively, drop the dough in rounded spoonfuls onto the prepared baking sheet. Bake 10-12 minutes, until cookies are set and slightly golden around the edges. Allow to cool on the cookie sheet before transferring to a wire rack.
To make the frosting, whisk all ingredients together in a bowl. Use a piping bag to make designs or simply spread on top of the cooled cookies.