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1
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 20 minutes. Drain potatoes and mash in the bottom of the pot until no lumps remain.
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2
Put pot over low heat. Stir butter into the mashed potatoes; cook, stirring constantly, until the butter has melted and the potatoes are no longer moist. Spread potatoes into the bottom of a wide bowl and refrigerate until cooled completely, at least 30 minutes.
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3
Beat egg yolks into cooled potatoes; season with salt and white pepper. Stir Cheddar cheese and diced onion into the potatoes. Shape potato mixture into tennis ball-sized spheres, arrange into a baking dish, and refrigerate until chilled completely through, at least 1 hour.
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4
Beat eggs and water together in a bowl. Spread flour and bread crumbs into two separate wide bowls.
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5
Gently roll potato balls in flour to coat and shake to remove excess flour. Dip each into the beaten egg, turning to coat. Press each egg-coated ball into bread crumbs, again turning to coat on all sides. Gently toss between your hands so any bread crumbs that haven't stuck can fall away. Place the breaded balls onto a plate while breading the rest; do not stack.
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6
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).
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7
Deep-fry potato balls in batches in hot oil until golden brown, about 5 minutes. Add more oil and return to 375 degrees F (190 degrees C) between batches. Drain fried balls on a plate lined with paper towel.
