In a 1½- to 2-quart saucepan, bring the broth and ½ onion to a simmer. If using raw chicken breasts, add them to the broth and poach gently until cooked through, about 5 minutes. Place the poached chicken (or chopped cooked chicken, if using) in a food processor with a steel knife blade. Pulse 2 or 3 times, until the chicken is finely chopped, but not pureed. Set aside. Strain the broth and set it aside; discard the onion.
In a large saucepan, melt the butter over medium heat. Add the minced onion and cook, stirring often, until softened and translucent, 3 to 4 minutes. Gradually stir in the ⅓ cup flour to make a paste. Whisk in ¾ cup of the reserved broth and the half-and-half. Cook, stirring, until the sauce is smooth and thick, 3 to 5 minutes. Add the nutmeg and season with salt and pepper. Fold in the chicken and the parsley, if using. Taste and adjust the seasonings. Spread the chicken mixture into a shallow baking pan and chill in the freezer or refrigerator until firm, about 30 minutes.
While the chicken mixture is chilling, in a mixing bowl, beat the eggs together with 1½ to 2 tablespoons water. Place the ¾ cup flour on a plate or small platter. Place the bread crumbs in a shallow baking dish. Shape heaping tablespoons of the chicken mixture into cylinders about ¾ inch thick and 2½ inches long. Dip the croquettes first in the flour, then in the eggs, and then roll them in the bread crumbs. As they are coated, place the croquettes on a baking sheet and refrigerate until all of the chicken mixture is used. Cover the croquettes and chill for at least 30 minutes to hold the shape and make the coating adhere. The croquettes may also be covered and frozen at this point for up to 2 weeks.
Preheat the oven to 250°F. In a deep-fryer or a deep, heavy skillet, heat at least 2 inches of oil to 360 to 375°F. Add the croquettes in small batches and fry, turning occasionally, until golden brown, 3 to 4 minutes. Remove them with a slotted spoon or spatula and drain or baking sheets lined with crumpled paper towels. To keep the croquettes warm as you fry them, move the baking sheet to the oven. Serve warm.