Put the ham and 1/2 cup milk in a small bowl; refrigerate 1 hour. Meanwhile, melt the butter in a medium saucepan over medium heat. Add the shallots, celery, garlic, thyme sprig, cayenne and 1/2 teaspoon each salt and black pepper. Cook, stirring, until the vegetables are soft, about 5 minutes; discard the thyme. Add the flour and cook, whisking, 1 minute. Whisk in the remaining 1 cup milk; cook, whisking, until thick, 2 minutes. Transfer to a bowl and refrigerate until cold, about 1 hour.
Transfer the chilled mixture to a food processor and pulse until almost smooth. Add the chicken and 1/2 teaspoon salt; pulse until the meat is finely ground. Drain the ham and add to the food processor along with the parsley; pulse until combined. Transfer to a bowl.
Brush a baking sheet with vegetable oil. With lightly oiled hands, shape the chicken mixture into about twenty 2-to-3-inch croquettes. Transfer to the baking sheet and refrigerate 30 minutes.
Spread the breadcrumbs in a shallow dish; whisk the eggs in another dish. Dip the croquettes in the eggs, then coat in the breadcrumbs; return to the baking sheet. Cover and chill until ready to fry.
Heat 2 inches of vegetable oil in a large pot over medium-high heat until a deep-fry thermometer registers 350 degrees F. Working in batches, fry the croquettes until browned, about 4 minutes. Drain on paper towels. Serve warm or at room temperature with honey mustard.
Photograph by Con Poulos
Recipe courtesy Food Network Magazine