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1
Heat the oil to 325 degrees F in a large, deep pot over medium heat.
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2
Regulate the temperature with a candy thermometer.
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3
Keep the potato sticks in a bowl of water to prevent browning before frying.
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4
Transfer to paper towels and dry well.
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5
Working in batches, fry the potatoes until they are cooked through and tender but have no color, 4 to 5 minutes.
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6
Be sure the fries have plenty of room and gently swish them around as they are cooking.
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7
Transfer immediately to a paper-towel-lined baking sheet.
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8
Raise the temperature of the oil to 375 degrees F.
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9
Heat the olive oil in a small nonstick skillet over medium heat.
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10
Fry the egg sunny-side up until the whites are just set, 3 to 4 minutes.
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11
Working in batches, fry the fries a second time at 375 degrees F, until they are golden brown and crispy.
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12
Transfer immediately to paper towels and toss with salt.
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13
Transfer to a serving platter, smother with the Fontina Cheese Sauce and top with the egg.
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14
Garnish with chives.
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15
Place the butter and onions in a large saucepan, sprinkling with salt, and cook until soft and aromatic, 8 to 10 minutes.
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16
Add the flour and cook, stirring frequently, until the mixture looks like wet sand, 4 to 5 minutes.
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17
Slowly whisk in the milk and season with salt.
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18
Bring to a boil, and then reduce to a simmer.
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19
Stir in the mustard and hot sauce and cook over low heat until the mixture is slightly thicker than heavy cream.
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20
Whisk in the fontina and Parmigiano-Reggiano to combine.
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21
Taste and adjust the seasoning with more mustard and hot sauce, if needed, adding a little more milk if the mixture seems too thick.
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22
Stir in the diced ham.
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23
The sauce should be very creamy and flavorful.
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24
Keep warm and reserve until the fries are ready.
