In a saucepan, melt the butter over medium heat. Add the flour and whisk constantly for about 2 minutes to create a smooth paste (roux).
Slowly pour in the milk, whisking continuously to prevent lumps. Cook over medium-low heat, stirring with a wooden spoon, until the béchamel thickens and coats the back of the spoon. This should take about 10-15 minutes.
Remove from heat and stir in the finely chopped jamón, a pinch of nutmeg, salt, and pepper. Spread the mixture on a flat plate or tray and cover it with plastic wrap, pressing it directly onto the surface to prevent a skin from forming. Let it chill in the refrigerator for at least 40 minutes, or until it's firm enough to handle.
Once the mixture is firm, use two spoons or your hands to shape it into small cylindrical or ball-shaped croquettes.
Prepare three shallow plates: one with flour, one with the beaten eggs, and one with breadcrumbs. Roll each croquette first in flour (shaking off any excess), then in the beaten egg, and finally in the breadcrumbs, making sure it is fully coated.
Heat the vegetable oil in a deep fryer or a deep pot to about 180°C (350°F).
Carefully place the croquettes in the hot oil in batches, making sure not to overcrowd the pan. Fry for 2-3 minutes, turning them occasionally, until they are golden brown and crispy.
Remove the croquettes with a slotted spoon and place them on a paper towel to drain excess oil. Serve immediately, while they are still hot and crispy.