Prepare the béchamel: In a medium saucepan over medium heat, melt 4 tablespoons of butter. Add the flour all at once and whisk constantly for 2 minutes to create a roux, cooking out the raw flour taste without browning.

Gradually add the milk in a slow stream while whisking constantly to prevent lumps. Continue whisking until the sauce thickens significantly, about 5-7 minutes. The mixture should be very thick, much thicker than a standard béchamel.

Remove from heat and stir in the finely chopped ham, nutmeg, black pepper, and a pinch of salt. Taste and adjust seasoning. The mixture should be well-seasoned as the breading will dilute the flavor.

Transfer the mixture to a shallow dish, spread it evenly, and cover with plastic wrap pressed directly onto the surface to prevent a skin from forming. Refrigerate for at least 3 hours or overnight until completely firm.

Once chilled, shape the mixture into small cylinders or oval shapes, about 2 inches long and 1 inch wide. Keep your hands lightly moistened to prevent sticking.

Set up a breading station: Place flour in one shallow dish, beaten eggs in a second dish, and breadcrumbs in a third dish.

Coat each croquette first in flour, shaking off excess, then dip in egg, and finally roll in breadcrumbs, pressing gently to adhere. For extra crunch, you can double-bread by repeating the egg and breadcrumb steps.

Heat vegetable oil in a deep skillet or pot to 350°F (175°C). The oil should be about 2 inches deep.

Fry the croquettes in batches, being careful not to crowd the pan, for 3-4 minutes, turning once, until deep golden brown on all sides.

Remove with a slotted spoon and drain on paper towels. Serve immediately while hot and crispy.