Beat shortening, sugars, and butter with a mixer on high speed until smooth, about 3 minutes. Add egg, and beat until mixture is pale and fluffy, about 2 minutes.

Reduce speed to low. Mix in flour mixture in 2 additions, alternating with milk and vanilla, beginning and ending with flour.

Drop mounds of dough (about 2 teaspoons each) onto parchment-lined baking sheets, spacing each 2 inches apart. Bake until cookies spring back when lightly touched with a finger, 12 to 14 minutes. Let cool on sheets set on wire racks for 10 minutes. Transfer cookies to racks, and let cool completely.