In a small saucepan, melt the butter over medium heat.
Add the flour and cook, stirring constantly, for approximately 3 minutes.
Gradually whisk in the milk, salt, pepper, and nutmeg.
Continue to cook, stirring constantly, until the mixture thickens and becomes smooth.
Spread the mixture onto a dinner plate and let it cool.
Refrigerate until it's firm enough to handle.
Shape the cheese mixture into 1¼-inch balls.
Place the beaten eggs and flour in separate shallow bowls.
Heat at least 1 inch of olive oil in a large skillet over medium-high heat, or better yet, use a deep fryer set at 365°F.
The oil should be hot enough that a cube of bread turns light brown in 60 seconds.
Dip each croquette in the beaten egg, dredge it in bread crumbs, and then place it in the hot oil.
Fry the croquettes, turning them several times, until they become golden.
Use a slotted spoon to transfer the fried cheese croquettes to paper towels to drain any excess oil.