Deep-Fried Creamy Chicken Gravy

1/2 cup unsalted butter
1/2 yellow onion, finely chopped
1/2 cup all-purpose flour
1/4 teaspoon freshly ground black pepper
1 pinch freshly grated nutmeg
1 pinch cayenne pepper
2 1/2 cups whole milk
1 teaspoon salt, plus more to taste
2 cups finely chopped, cooked chicken meat
1/2 cup finely chopped ham
2 tablespoons freshly chopped Italian parsley
Breading:
1 cup all-purpose flour, or as needed for dredging
2 eggs, beaten
2 cups bread crumbs
2 cups vegetable oil for frying

Melt butter in pan over medium heat. Add onions; cook and stir until onions begin to turn translucent, about 5 minutes. Stir in 1/2 cup flour and cook until flour loses its raw taste, 3 or 4 minutes. Add black pepper, nutmeg, and cayenne pepper; stir. Remove from heat.
Pour milk into butter-onion mixture to make the bechamel sauce. Whisk until incorporated. Return pan to medium-high heat and continue whisking until mixture thickens, 1 or 2 minutes; continue to cook and stir 3 or 4 minutes. Remove from heat and stir in salt.
Place chopped chicken, ham, and parsley in bowl. Add bechamel sauce and mix thoroughly. Allow mixture to cool to room temperature; cover with plastic wrap and refrigerate until firm, 2 or 3 hours.
Heat oil in a deep fryer or large saucepan to 350 degrees F (175 degrees C).
Place 1 cup flour, beaten eggs, and bread crumbs in 3 separate shallow bowls.
Scoop out chilled mixture into evenly-sized portions (about the size of a golf ball) to form croquettes. Roll balls first in flour, then in beaten eggs, and then in bread crumbs.
Fry croquettes in batches until crispy, about 4 minutes. Drain on paper towels and transfer to rack.

Notes
These came out a lot creamier than what I wanted, so these chicken croquettes are more like deep-fried chicken gravy.
Still, they're delicious. If you reduce the amount of butter and milk, you'll come closer to what I was after.

Chef John