How to Make It
Whisk together flour and salt in a medium bowl; set aside.
Beat together sugar, butter, oil, and 1 1/2 teaspoons vanilla in a large bowl with an electric mixer on medium speed until light and fluffy, about 3 minutes. Beat in egg. Add flour mixture; beat on low speed just until combined. Divide dough in half. Roll each half between 2 sheets of plastic wrap to 1/4-inch thickness. Place dough halves on a baking sheet, and freeze until firm, about 20 to 30 minutes.
Preheat oven to 350°F. Remove plastic wrap from 1 dough half. Working quickly, cut dough with a 2 1/2-inch cutter of desired shape to form 16 cookies, rerolling scraps as necessary. (Refreeze dough if it gets too soft.) Repeat procedure with remaining dough half. Arrange cookies on 2 baking sheets lined with parchment paper. Bake at 350°F until lightly browned around edges, about 10 minutes. Cool on baking sheets set on a wire rack for 2 minutes. Transfer cookies directly to wire rack to cool completely, about 30 minutes.
Place powdered sugar, 1 1/2 tablespoons water, meringue powder, and remaining 1/4 teaspoon vanilla in a medium bowl. Beat with an electric mixer on low speed just until moistened, about 20 seconds; beat on high speed until thick, bright white, and smooth, about 5 minutes. Beat in additional water, 1/2 teaspoon at a time, if needed, to reach desired consistency. Stir in food coloring, if desired. Spread or pipe icing evenly over cookies. Let stand at room temperature until icing is set, 1 to 2 hours. Store in an airtight container up to 3 days.