Boil potatoes: Place potato quarters in a large pot, cover with cold water, and bring to a boil. Cook until very tender (easily pierced with a fork), about 15-20 minutes.

Drain and mash: Drain potatoes thoroughly. Return to the pot and mash until completely smooth and creamy. Use a potato masher, hand mixer, or electric mixer.

Cook sugar and butter: In a saucepan, combine sugar and butter over medium heat. Stir constantly until butter melts and sugar dissolves. Bring to a gentle simmer, stirring frequently to prevent burning.

Combine and stir: Pour the sugar-butter mixture into the mashed potatoes. Stir constantly until well combined and smooth. The mixture will thicken slightly.

Add flavorings: Stir in vanilla and salt. Add optional nuts and marshmallows, stirring gently to incorporate.

Shape and cool: Grease a baking sheet with butter. Drop spoonfuls of the mixture onto the sheet, shaping into small mounds or squares, leaving space between each.

Cool completely: Let the potato candy cool completely at room temperature before serving. This allows them to firm up.