1Melt butter in a large skillet over medium heat. Stir in onion, green onions, and garlic; cook until tender, about 10 minutes.
Transfer contents of skillet to a large bowl. Stir in ham, Cheddar cheese, and 1 1/2 cups of bread crumbs. Beat 3 eggs; stir into ham mixture. Sprinkle with salt and pepper. Using your hands, make golf ball-size balls out of the ham mixture.
Heat vegetable oil in a large skillet over medium heat. Spread remaining bread crumbs onto a plate. In a large bowl, beat 1 egg; stir in water. Dip balls into egg mixture, and roll in bread crumbs. Place balls in hot oil being careful not to crowd. Fry until golden brown. Remove to paper towels to soak up grease.