Place the cubed potatoes in a pot of salted water. Bring to a boil and cook until the potatoes are fork-tender, about 15 minutes. Drain the potatoes and transfer them to a mixing bowl.

Mash the cooked potatoes until smooth, then add the shredded cheddar cheese, grated Parmesan cheese, green onions, garlic powder, salt, and pepper. Mix until well combined.

Take a small portion of the potato mixture and shape it into a cylinder about 2 inches long. Repeat this process until all the mixture is used, yielding approximately 12 croquettes.

Set up a breading station with three shallow dishes: one with the all-purpose flour, one with the beaten eggs, and one with the breadcrumbs.

Roll each croquette in the flour, dip it in the beaten eggs, and then coat it thoroughly with breadcrumbs. Place the breaded croquettes on a plate and refrigerate for 15-20 minutes to firm up.

In a deep saucepan or frying pan, heat vegetable oil to 350°F (175°C). Carefully lower the croquettes into the hot oil in batches and fry until golden brown and crispy, about 3-4 minutes.

Using a slotted spoon, transfer the fried croquettes to a plate lined with paper towels to drain excess oil.

Serve the cheesy potato croquettes hot with your favorite dipping sauce or as a side dish to complement your meal.