In a large bowl, beat together the butter and sugar until light and blended, about 2 minutes. Beat in egg and vanilla. On low speed, beat in flour until dough begins to come together. Gather dough into a ball, flatten into a disc and refrigerate 2 hours or overnight.
Heat oven to 350 degrees F. Roll out dough on lightly floured surface to 3/16-inch thickness. Cut out circles using 3-1/2-inch and 2-1/2-inch cutters. Transfer circles to ungreased cookie sheets. Reroll scraps and cut more circles. Bake cookies at 350 degrees F. for 14 to 16 minutes, rotating halfway, until just browning at edges. Transfer cookies to a rack to cool.
Combine confectioners sugar, powdered egg whites and 7 tablespoons water in a large bowl. With electric mixer on low speed, beat until blended then increase speed to high and beat 6 minutes. Remove 1 cup icing and tint black with food coloring. Tint remaining icing with yellow food coloring. Mix until smooth. Spread yellow icing onto cookies using a small angled spatula. Let dry completely.
Transfer black icing to a resealable plastic bag or a piping bag fitted with a writing tip. Pipe eyes and a smiling mouth onto each cookie. Let dry completely at room temperature before stacking cookies. Makes ten 3-1/2- inch cookies and twelve 1-1/2- inch cookies.