In a stand mixer, beat together the butter, sugar, salt, and mashed potatoes until smooth. Add eggs and yeast mixture, and stir to combine.
Add the flour, 1 cup at a time, until a soft dough forms.
Place the dough in a greased bowl, and cover with plastic wrap. Let sit until doubled in size, about 1–2 hours depending on how warm your kitchen is.
Punch dough down and divide into thirds. Take one third and divide the dough into 12 balls. Arrange the 12 balls in one of the 9-inch cake pans. Repeat with the remaining dough.
Cover with plastic wrap (spray the plastic wrap with cooking spray first) and let sit until doubled in size, about 1 hour.
Preheat oven to 375°F. Bake for 20–22 minutes, until lightly golden brown.
While the rolls are baking, melt the salted butter. When the rolls are done, brush the melted butter over the warm rolls.
Notes
You can use regular mashed potatoes or instant mashed potatoes (made according to package directions) for this recipe. Avoid skin-on mashed potatoes. To freeze before baking, shape the dough into balls after the initial rise. Place the dough balls in a single layer on a baking sheet that fits in your freezer, and freeze until solid. Remove them from the baking sheet and place in a resealable freezer bag. Freeze for up to 2 months.To bake, place the balls in a greased cake pan (up to 12 per pan), cover with greased plastic wrap, and allow to rise for 3–4 hours. Bake as directed.To freeze after baking, remove the rolls from the pan and wrap tightly in aluminum foil. Transfer to a resealable freezer bag and freeze for up to 2 months.Defrost overnight at room temperature and heat in the oven for 5–8 minutes (while still wrapped in foil) until warm.