Melt butter in a small skillet over medium heat. Add garlic and dried onion; cook and stir until browned, about 5 minutes. Pour mixture into the bowl with the ham; add the beaten eggs, 1 cup bread crumbs, Cheddar cheese, brown sugar, and black pepper. Mix by hand until combined; mold into 15 to 18 medium croquettes, or rectangular balls.
Beat remaining 1 egg in a bowl with 1 tablespoon water. Pour remaining 2 cups bread crumbs into a large nonstick baking pan. Dip each croquette in the egg and roll in bread crumbs.
Heat olive oil in a large skillet over medium-high heat. Add croquettes; fry in batches until golden brown, about 10 minutes total. Place finished croquettes on a paper towels to drain excess oil.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.