Combine butter, granulated sugar, and vanilla in mixer bowl. Beat on medium speed until light and fluffy, about 3 minutes. Add egg and egg yolk; beat on low speed until blended. Gradually add flour and salt, beating just until blended.
Refrigerate dough, wrapped in plastic wrap, at least 2 hours or up to 2 days.
Heat oven to 350°F. Work with 1/2 of the dough at a time, keeping remaining dough refrigerated. Roll out dough on a lightly floured surface to 1/4-inch thickness. Cut out cookies using cookie cutters; place 1 inch apart on parchment paper-lined or ungreased baking sheets.
Bake in 350°F oven until edges are lightly browned, 8 to 10 minutes. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.
For the icing: combine powdered sugar, meringue powder, and water in mixer bowl. Beat on high speed until light and fluffy, 6 to 8 minutes. If icing is too thick, thin with a small amount of additional water.