Preheat oven to 325°. Combine butter, sugar, vanilla and salt, and beat with electric mixer on medium until blended. Add flour; mix on low until incorporated. Shape dough into a disk and wrap in wax paper. Chill for 2 hours.
Soften dough at room temperature. Place oven racks in upper and lower thirds of oven.
Halve dough. Roll to 1/4 inch thick and cut out shamrock shapes; save scraps. Repeat. Place cookies on two parchment-paper-lined baking sheets. Bake for 20 minutes (switch oven racks and turn pans halfway through cooking) until lightly golden. Remove; top with sprinkles. Transfer to cooling rack.
Mold scraps into 1-inch balls; roll balls in sugar. Arrange 2 inches apart on baking sheet; flatten to 1/4 inch thick with bottom of a glass. Bake for 20 minutes on top rack. Transfer to cooling rack.