Baked Italian Potato Croquettes

Add the cut potatoes to a large pot filled with cold water; cover the pot and bring to a boil. Cook the potatoes for 10-15 minutes or until fork tender.
Drain the potatoes well and place them in a large bowl. Using a potato masher, mash the potatoes until smooth. Stir in the Pecorino, salt, pepper, and parsley, if using. Fully incorporate but don't overmix. Place in the refrigerator, covered, for about 30 minutes or until cool enough to handle.
Preheat the oven to 400F. Pour the 2 tablespoons of olive oil in the bottom of an 11x14" pan.
Remove the potatoes from the fridge, and stir in the 2 eggs just until incorporated
For the Panko coating, beat the third egg in a small bowl along with the 1 tablespoon of water. Place the Panko breadcrumbs in another bowl, and toss them with the 1 tablespoon of olive oil.
Scoop out some of the potato mixture and form it into an approximately 3" round disc. Quickly toss it in the beaten egg, coat it with Panko, and place it onto a large plate or platter. You can also baste both sides of the croquette with a basting brush if that's easier. Repeat this process until you've used up all the potato mixture.
Place 8 croquettes in the pan, and refrigerate the others until they're ready to bake. Bake on the bottom rack of the oven for 12-15 minutes, until golden brown on one side. Flip them, and cook the other side for another 12-15 minutes. You may need to add more oil to the pan when you flip them.