Melt the butter in a saucepan over medium heat.
Whisk in the flour and cook for 2-3 minutes, stirring constantly, until the mixture forms a thick paste.
Gradually pour in the milk, whisking continuously to prevent lumps from forming.
Bring the mixture to a simmer and cook, stirring frequently, until thickened, about 10-15 minutes.
Remove from heat and stir in the nutmeg, salt, and pepper to taste.
Fold in the chopped ham.
Spread the mixture into a baking dish and refrigerate for at least 2 hours, or until firm.
Scoop the chilled mixture by the tablespoonful and roll into oval shapes.
Dip the croquettes first in the beaten eggs, then in the breadcrumbs, coating them completely.
In a large skillet, heat 1-2 inches of vegetable oil over medium-high heat.
Fry the croquettes in batches until golden brown on all sides, about 2-3 minutes per batch.
Drain on a paper towel-lined plate and serve hot.
