Wash and peel the potatoes. Transfer to a large pot and cover with cold water, then turn on the heat and cook until they are tender, but not too soft. After cooking, drain and let cool.
Wash the zucchini and the carrots, then peel the carrots. Transfer to a pot with water and cook for a couple of minutes. Do not cook for too long, drain the vegetables when they are still firm.
Once they have cooled, mash the potatoes and place in a large mixing bowl.
Dice the zucchini and the carrots into small cubes, then transfer to the bowl with mashed potatoes. Mix just until combined, making sure you do not mash the vegetables. Add onion powder, garlic powder ,salt and pepper, adjusting to your taste preferences.
Add egg, olive oil, and 1 to 2 tablespoons breadcrumbs. Give it a quick stir until you get a good dough, adding more breadcrumbs if it is necessary.
Shape the veggie nuggets, then place over a parchment paper covered baking dish, in the air fryer basket, or in a plate. If desired, you can also coat the nuggets in more breadcrumbs to make them crispier.