Begin by boiling around 2 pounds of peeled and chopped potatoes in a large pot of salted water. Cook for 15-20 minutes until fork-tender. Drain and mash until smooth; set aside to cool.

In a mixing bowl, combine the cooled mashed potatoes, half of the shredded cheese, chopped green onions, salt, pepper, and the egg. Stir until well incorporated.

Scoop approximately one tablespoon of the potato mixture and shape it into a ball. Repeat this process until all the mixture is formed. Place on a parchment-lined baking sheet.

In a shallow bowl, pour in breadcrumbs. Roll each puff in breadcrumbs to coat thoroughly. Arrange back on the baking sheet.

For frying, heat about 1 inch of vegetable oil in a skillet until hot, around 350°F. Fry puffs for 4-5 minutes until golden brown. For baking, preheat oven to 400°F, spray puffs with cooking spray, and bake for 20 minutes.

Allow puffs to cool for a few minutes on a wire rack. Serve warm with dipping sauces.