In the bowl of a stand mixer, cream together pudding, butter, and granulated sugar until fluffy.

Scrape down sides and beat in egg. Turn the speed down to low and slowly mix in flour and salt until well-combined.

Spread dough onto a sheet of parchment paper. Cover with another sheet of parchment, and roll dough to approximately 1/8-inch thick. Transfer dough (on the parchment) to a cookie sheet and freeze for 20 minutes.

Once dough has chilled, remove from the freezer and place parchment on counter. Remove the top parchment, flip it over, place it on the dough again, and flip the entire thing over. Remove the parchment that is now on top. (This step will help keep your cookies from sticking to the parchment after cutting.) Cut out rounds using a 3-inch round cutter. For half the cookies, cut out the center shape with a 1.5-inch clover cutter. Place cookies back on baking sheet and put in the freezer.

Preheat oven to 350°F. Once the oven has preheated, remove cookies from the freezer, transfer to a clean parchment-lined baking sheet, and bake cookies for 6-8 minutes.

Let cookies cool on baking sheet for 2 minutes before removing to a wire rack.

Once cookies are completely cooled, make the frosting. In the bowl of a stand mixer, beat together softened butter and powdered sugar. Add the vanilla paste and milk and beat until frosting is fluffy.

To fill the cookies, fit a pastry bag with an M2 tip and fill bag with frosting. Flip over full cookie round and pipe frosting onto bottom, starting from the outside and working your way in, in a circular motion. Top frosted cookie with clover-hole cookie.

To add sprinkles, fill clover hole completely with sprinkles. Gently press on sprinkles so they adhere evenly, then tip cookie onto a rimmed plate to remove excess sprinkles.