Remove crust and slice bread into 1/2-inch cubes. Heat butter and oil together in a skillet over medium heat; fry bread cubes until golden brown, 3 to 5 minutes. Transfer to a paper-towel lined plate.
Bring a large pot of salted water to a boil. Add potatoes; cook until tender, about 45 minutes. Drain; allow to cool slightly. Peel and cut into large pieces. Transfer to a bowl; refrigerate about 30 minutes.
Mash potatoes with a fork; stir in salt and nutmeg. Add flour and cornstarch. Knead mixture in bowl, using your hands, until a smooth dough forms. Add more flour by tablespoon if dough is sticky. Mix in egg.
Roll dough into balls, 1/4 cup per dumpling. Insert a bread cube into the center of each dumpling; roll between palms to smooth over.
Bring a large pot of salted water nearly to a boil. Place 4 dumplings at a time into the pot; cook until they float to the surface, 10 to 15 minutes. Transfer cooked dumplings to a large bowl using a slotted spoon; keep warm. Repeat with remaining dumplings.