Beat butter in a bowl with an electric mixer until creamy. Add 1 1/2 cups confectioners' sugar gradually, beating until light and fluffy. Beat in egg and 1 teaspoon vanilla extract.
Combine flour, baking soda, cream of tartar, and salt in a second bowl. Add flour mixture to the creamed butter mixture and mix to combine. Wrap in plastic wrap and chill in the refrigerator for at least 1 hour.
Preheat oven to 400 degrees F (200 degrees C).
Dust a work surface with flour and roll out dough into a thin circle. Cut out tulip shapes; if your tulip cookie cutter has a stem, cut the stem off with a knife, so you have a stemless tulip shape. Arrange cut-out cookies on ungreased baking sheets.
Bake in the preheated oven until lightly browned, 6 to 8 minutes. Remove from baking sheets carefully and transfer to wire racks. Cool completely, about 15 minutes.
Stir 1 cup confectioners' sugar and milk together in a small bowl until smooth. Beat in corn syrup and 1/4 teaspoon vanilla extract until icing is smooth and glossy. Add more corn syrup if icing is too thick.
Spoon icing into a piping bag with a small plain tip. Pipe icing around the edge of each cookie to create a border. Fill in the middle completely with icing. Stick 2 chocolate chip cookies into the wet icing for the eyes. Let cookies stand until icing dries, about 1 hour.