Place potatoes in a saucepan and cover with water. Bring to a gentle boil over high heat; cook until tender and drain.
In a food processor with chopping blade, pulse corned beef until coarsely chopped. Alternatively, shred by hand.
Heat oil over medium heat in a large cast-iron or nonstick skillet. Add onion and garlic and saute. Add potatoes, salt, and pepper. Reduce heat to low. Add the chopped corned beef, mustard, and butter. Cook until browned on one side. Serve with cooked eggs.