In a medium saucepan, add the potatoes and enough cold water to cover. Add salt and bring to a boil. Lower the heat and simmer until tender, about 8 minutes. Drain in a colander and let cool.
In a heavy skillet, heat 1 tablespoon each olive oil and butter over medium heat. Add the mushrooms, onion and herbes de Provence; season with salt and pepper and cook, stirring often, until the vegetables are tender, about 10 minutes. Transfer to a bowl and cover to keep warm. Wipe out the skillet.
Add the remaining 1 tablespoon each olive oil and butter to the skillet and heat over medium heat. Add the potatoes, season with salt and pepper and fry, without disturbing, until crusty, about 5 minutes. Flip and fry until golden on the other side, about 3 minutes more. Stir in the mushrooms and cook until heated through.
Meanwhile, fill a large, deep skillet with 1 1/2 inches of water. Add a splash of vinegar and a pinch of salt, cover and bring to a simmer. Working in 2 batches, crack 4 eggs into the simmering water and cook until desired doneness, 2 to 3 minutes for soft yolks. Using a slotted spoon, transfer the eggs to paper towels and blot dry slightly.
Divide the home fries among 4 plates, top with 2 eggs each and season with salt and pepper.