Heat the oil to 325 degrees F in a large, deep pot over medium heat. Regulate the temperature with a candy thermometer.

Keep the potato sticks in a bowl of water to prevent browning before frying. Transfer to paper towels and dry well. Working in batches, fry the potatoes until they are cooked through and tender but have no color, 4 to 5 minutes. Be sure the fries have plenty of room and gently swish them around as they are cooking. Transfer immediately to a paper-towel-lined baking sheet.

Raise the temperature of the oil to 375 degrees F.

Heat the olive oil in a small nonstick skillet over medium heat. Fry the egg sunny-side up until the whites are just set, 3 to 4 minutes.

Working in batches, fry the fries a second time at 375 degrees F, until they are golden brown and crispy. Transfer immediately to paper towels and toss with salt. Transfer to a serving platter, smother with the Fontina Cheese Sauce and top with the egg. Garnish with chives.

Place the butter and onions in a large saucepan, sprinkling with salt, and cook until soft and aromatic, 8 to 10 minutes. Add the flour and cook, stirring frequently, until the mixture looks like wet sand, 4 to 5 minutes. Slowly whisk in the milk and season with salt. Bring to a boil, and then reduce to a simmer. Stir in the mustard and hot sauce and cook over low heat until the mixture is slightly thicker than heavy cream. Whisk in the fontina and Parmigiano-Reggiano to combine. Taste and adjust the seasoning with more mustard and hot sauce, if needed, adding a little more milk if the mixture seems too thick. Stir in the diced ham. The sauce should be very creamy and flavorful.

Keep warm and reserve until the fries are ready.