Preheat the oven to 375 degrees F (190 degrees C).
Place turnips into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and return to the pot. Mash turnips; stir in butter and eggs.
Combine flour, brown sugar, baking powder, salt, pepper, and nutmeg in a bowl and stir into mashed turnip mixture. Transfer mixture to a baking dish.
Mix bread crumbs and melted butter for topping in a small bowl; sprinkle over turnip mixture.
Bake in the preheated oven until light brown on top, about 25 minutes.