Put the potatoes in a large pot and cover with cold water. Bring to a boil over medium heat and cook until the potatoes are tender, about 15 to 20 minutes. Drain the potatoes and return to the pot over low heat and let the potatoes dry for 3 minutes. Mash the potatoes with a potato masher.
In a small pot, over low heat, combine the cream, butter, and salt and pepper, to taste. Heat until the butter is melted and the cream is hot. Slowly add the heated cream and butter to the potatoes and mix in with a wooden spoon or a hand mixer until all the cream is incorporated. Be careful not to overmix or the potatoes will become gummy.
Spread the potatoes out onto a sheet pan to cool. Once the potatoes have cooled to room temperature scoop out into 1 1/2 tablespoon portions. Roll each portion into a ball and then flatten into a patty about 1 1/2 inches in diameter. Arrange the patties in 1 layer on sheet trays and refrigerate for 1 hour until cold.
Line a sheet tray with 1 cup bread crumbs. Line another sheet tray with paper towels. Reserve both.
Set up a dredging station with 3 baking dishes. In 1 baking dish combine the flour, cayenne pepper, and season with salt and pepper. In the second baking dish beat together the eggs with the milk. In the third baking dish mix 2 cups bread crumbs with salt and pepper, to taste. Dredge each cake into the seasoned flour then into the egg mixture and then into the bread crumbs making sure they are completely coated. Arrange the bread cakes on the sheet tray with the bread crumbs on the bottom.
Heat about 1/2-inch of oil in large saute pan over medium-high heat. Fry the cakes until golden brown in batches, about 3 to 4 minutes per side. Transfer the cakes as they are fried to the baking sheet lined with a paper towel to drain. Blot the tops of the cakes with paper towels to remove the excess oil. Transfer the cakes to a serving platter.
Melt the butter in a large skillet over low heat, whisk in the flour and cook until flour turns a light brown color, about 3 to 4 minutes. Slowly whisk in the chicken broth. Add the hot sauce, Worcestershire sauce, apple butter, thyme leaves, and season with salt and pepper. Let simmer until thick, about 5 minutes. Remove the sauce from pan to a serving bowl and serve on the side as a dipping sauce with the mashed potato cakes.