Melt 1 tablespoon butter in a large skillet over medium heat. Add minced onion and garlic, cooking until softened, about 3 minutes. Add chopped jamón and cook 2 minutes more. Set aside.

In a large saucepan, melt remaining 5 tablespoons butter over medium heat. Add flour and whisk vigorously to create a roux. Cook 2-3 minutes, stirring constantly, until slightly golden.

Gradually add milk in a slow stream while whisking constantly. Continue whisking until béchamel thickens considerably, 8-10 minutes. Mixture should be very thick.

Stir in jamón mixture, nutmeg, salt, and pepper. Mix thoroughly until evenly distributed.

Transfer mixture to a greased baking dish, spreading to 1-inch thickness. Cover with plastic wrap pressed onto surface. Refrigerate 4 hours or overnight.

Once chilled, portion mixture into small amounts and roll into oval shapes about 2 inches long.

Set up breading station with beaten eggs and breadcrumbs. Dip each croquette in egg, then breadcrumbs. Repeat for double coating if desired.

Heat oil to 350°F in a deep pot. Fry croquettes in batches of 4-5, turning occasionally, until deep golden brown, 3-4 minutes.

Drain on paper towels and serve immediately while hot and crispy.