In using a stand mixer or hand 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.
Dump 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 chill for 10 minutes.
Once dough has chilled, place cookies and parchment on counter. Remove the top parchment, flip it over, place it on the dough again, and carefully 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. Note that clover cut-outs will bake faster than larger cookies.
Let cookies cool on baking sheet for 2 minutes before removing to a wire rack to cool completely.
To make the frosting, beat together softened butter and powdered sugar. Add the vanilla and milk and beat until frosting is fluffy.
Spread frosting on bottom of full cookie round in a circular motion. Top with clover-hole cookie and press gently. Serve.
Notes
Dough might seem crumbly at first but will come together while rolling.
If dough is too cold to cut, let rest a few minutes to warm up.
You can bake the clover cut-outs as they are, or re-roll the dough to create more sandwich cookies.
If you want a decorate edge for the frosting, use a pastry bag and M2 tip, then pipe from the outside to the middle in a circle.
For extra festivity, add sprinkles to the center hole.