In a large saucepan filled with salted cold water, bring potatoes to a boil; reduce heat and simmer 20 minutes, or until tender. Drain in colander, transfer to mixer bowl, and beat on low speed with a paddle attachment until smooth.
Meanwhile, in a medium saucepan, simmer chestnuts in heavy cream, about 15 minutes, until very soft. Add chestnut mixture to potatoes and beat on medium speed until combined. Season to taste with salt and pepper. Let cool to room temperature.
Using a 1/4-cup measuring cup, form potato mixture into 1-inch-thick patties.
Dredge potato cakes in flour, then egg, and then in bread crumbs. Place on a baking sheet and refrigerate, covered, 1 to 4 hours.
In a large skillet, heat oil to 375°F. Fry potato cakes 2 minutes on each side or until golden brown. Drain on paper towels. Serve hot.