In a large mixing bowl, combine the shredded chicken, diced ham, breadcrumbs, grated cheese, and green onions. Mix well to evenly distribute the ingredients.

In a separate small bowl, combine the mayonnaise, Dijon mustard, garlic powder, onion powder, salt, and pepper. Stir until smooth.

Add the mayonnaise mixture to the meat and breadcrumb mixture. Mix thoroughly until all ingredients are well combined.

Shape the mixture into small patties or croquettes, about 2-3 inches in diameter. Place them on a baking sheet lined with parchment paper.

Set up a breading station: Place the flour in a shallow dish, the beaten eggs in another dish, and a plate nearby for the breaded croquettes.

Dredge each croquette in flour, shaking off any excess. Dip it into the beaten eggs, allowing the excess to drip off, and then coat it with additional breadcrumbs.

Heat vegetable oil in a large skillet over medium heat. Once the oil is hot, carefully place the croquettes in the skillet, making sure not to overcrowd the pan.

Fry the croquettes for about 3-4 minutes on each side or until they are golden brown and crispy. Remove them from the skillet and drain on paper towels.

Serve the ham and chicken croquettes warm with your favorite dipping sauce or condiment.