Cream the butter and sugar until fluffy. Beat in the egg and vanilla.
Whisk the flour, baking soda, and salt; mix into the wet ingredients. Chill the dough 1 hour.
Preheat the oven to 350°F. Roll the dough 1/4 inch thick and cut 3-inch rounds. Bake 9 to 11 minutes, until just set. Cool completely.
Whisk the icing ingredients to a thick but spreadable consistency. Spread a "melted puddle" circle of white icing on each cookie.
Press in mini chocolate chips for the eyes and coal smile, and an orange jimmy slightly off-center as the carrot nose. Let the icing harden.