Gather the ingredients.

Heat ½ cup olive oil in a large skillet over medium heat. Add the flour and cook for 3 minutes, stirring constantly, until a smooth paste forms.

Very slowly drizzle in the milk and chicken broth, stirring constantly, until well combined. This can take up to 30 minutes.

Add the nutmeg, and salt, and pepper to taste. Cook over medium heat, stirring constantly until the mixture is thick, smooth, with a consistency similar to mashed potatoes.

Lower the heat to low. Add the minced ham, stirring constantly until well combined, 2 to 3 minutes. Remove from heat. Adjust seasoning if necessary.

Let cool, 5 to 10 minutes. Refrigerate until the mixture is cold, about 3 hours. If you are preparing a day or so ahead, you can cover the mixture tightly and keep refrigerated until you are ready to fry the croquettes.

Beat the eggs with the water in a separate small, shallow bowl.

Pour the breadcrumbs into another small, shallow bowl.

Cover your hands in flour, then divide the mixture into 1-inch balls or oblong pieces. Set on a plate so that they are not touching.

Add about ½-inch oil to a large skillet or other heavy-duty pan and heat to 355 F.

Dip the croquettes in the egg, then coat with breadcrumbs.

Fry the croquettes in batches, turning several times, until golden.

Remove the croquettes with a slotted spoon to a paper-towel-lined plate. Serve immediately.