Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 15 minutes. Drain and mash. Set aside.

Preheat the oven to 450 degrees F (230 degrees C). Butter 24 mini muffin cups.

Combine water and butter in a saucepan over medium heat and season with salt. Heat until butter melts and starts to bubble. Stir in flour until mixture pulls away from the pan and comes together into a dough.

Transfer pastry dough to a bowl. Spread in the bowl and let cool, about 5 minutes. Mix in eggs using a spatula until dough is very soft and sticky. Add 2 cups of the mashed potatoes and mix until well combined. Add a pinch of salt, cayenne pepper, and nutmeg. Give it one last mix.

Scoop potato dough into the greased muffin cups.

Bake in the preheated oven until golden brown and puffed, about 20 minutes. Let rest for 1 minute before removing puffs onto a wire rack. Let cool for 5 minutes and serve warm.