Boil the potatoes on high heat until very tender, approx 20 minutes.
Drain very well.
Using a masher and a lot of elbow grease, a ricer, or something creative like my masticating juicer, get the potatoes as smooth as possible. Bonus points for squishing the mash through a fine mech sieve.
Add 1/2 the melted butter, and everything other than the eggs, and stir until well combined.
Once cool, slowly stir in the egg yolks. We do not want to make scrambled eggs in our potatoes.
Transfer to a piping bag fitted with a large star tip.
Pipe rounds approx 2.5" wide and 2" tall on a parchment paper-lined baking sheet.
Carefully brush with the remaining melted butter.
Bake at 425F or until the edges are golden brown.