To make the baked vegetable patties, wash the potatoes thoroughly and place them in a saucepan covered with water. Bring to a boil and cook until tender. Wash the zucchini and carrots, and peel the carrots. Boil them separately, draining them while they are still firm. Peel the cooked potatoes and pass them through a potato ricer into a bowl. Allow them to cool. Trim the zucchini and carrots, then cut them into small dice.
Add the carrots and zucchini to the mashed potatoes, add the chopped basil, and season with salt to taste. Mix the ingredients well to combine without mashing the diced vegetables.
Shape the mixture into portions roughly the size of a walnut. Bread them by dipping first into the beaten eggs and then into the breadcrumbs, then arrange them on a baking sheet lined with parchment paper. Bake in a preheated oven at 200°C (400°F) for about 30 minutes, then brown under the broiler for an additional 5 minutes. Remove from the oven and let them rest for a few minutes.
Transfer the vegetable patties to a serving platter, garnish with a few fresh basil leaves, and serve warm or at room temperature.