Mix the ground beef, breadcrumbs, egg, onion, 1/4 cup milk, nutmeg, salt, and pepper in a bowl until just combined. Form the mixture into 1-inch meatballs.
Melt 2 tablespoons of butter in a large skillet over medium heat. Add the meatballs and cook until browned on all sides, then remove them to a plate.
Add the remaining 2 tablespoons of butter to the skillet. Stir in the flour and cook for 1 minute. Slowly whisk in the beef broth and 1/2 cup milk until smooth.
Return the meatballs to the skillet and simmer for 10 minutes until the gravy thickens and the meatballs are cooked through.
While the meatballs simmer, place the potatoes in a saucepan and cover with water. Bring to a boil and cook until tender, about 15 minutes. Drain the potatoes.
Mash the potatoes with 1/4 cup milk and a pinch of salt until smooth. Serve the meatballs and gravy over the mashed potatoes.