Place unpeeled potatoes in a large pot and cover with cold water. Generously season the water with salt and bring it to a boil. Cook until potatoes are fork-tender. Do this for 25-35 minutes, depending on size.
Drain potatoes and cool slightly. When cool enough to handle, peel the potatoes. Place potatoes through a potato ricer into a large bowl and make sure to mash it well, especially the large chunks.
Add egg yolk. Mix and add fresh parsley, parmesan, and cubed mozzarella. Mix well and shape into 8 oval croquettes. Arrange the croquettes on a small baking sheet lined with parchment paper. Place in the refrigerator for 30 minutes.
In a large saucepan, heat vegetable oil to 350˚F.
Prepare a breading station: In one shallow dish, beat two large eggs. Place the bread crumbs in a second shallow dish.
Generously, season the beaten egg mixture and the breadcrumbs with a generous pinch of salt and pepper.
Roll each croquette into the beaten eggs. Let excess egg drip off and then dredge in the breadcrumbs, pressing lightly so the breadcrumbs will stick to the croquettes.
Fry the croquettes, a few at a time, in the hot oil until golden brown and crisp on all sides. Do this for approximately 3-4 minutes. Avoid crowding the oil and allow it to return to 350˚F before adding the next batch of croquettes.
Remove from oil as they are ready and place on paper towels or a wire rack. Serve immediately with desired dipping sauces. Enjoy!