Melt butter in a large saucepan over medium heat. Add onion, celery and carrot and cook, stirring, until vegetables have softened, about 5 minutes. Turn heat to high, add beef and cook, breaking up with a spoon, until no longer pink, about 5 minutes. Season with salt, pepper and nutmeg.
Pour in milk and wine and bring to a boil. Lower heat and cook at a lively simmer until most of liquid has evaporated, about 15 minutes.
Transfer mixture to slow cooker, stir in tomatoes, cover and cook on low for 8 hours. Remove cover, stir and cook until sauce is slightly thickened, about 30 minutes longer.