Place the quartered Yukon Gold potatoes in a large pot and cover with water. Add a pinch of salt. Bring to a boil over medium-high heat and cook until the potatoes are fork-tender, about 15–20 minutes. Drain the potatoes well.

Return the drained potatoes to the pot or a bowl. Add 2 1/2 tablespoons unsalted butter and salt and pepper to taste. Mash until smooth. Let the mashed potatoes cool to room temperature.

When the potatoes are cool, add 2 egg yolks, 1/2 cup grated Parmesan cheese, and 2 tablespoons chopped fresh parsley. Mix until just combined.

With clean hands, form the potato mixture into small logs or oval croquettes. Place the formed croquettes on a plate or baking sheet as you work.

Set up three shallow dishes: one with 4 tablespoons all-purpose flour, one with the 1 large beaten egg (egg wash), and one with 1 cup whole wheat breadcrumbs.

Lightly coat each croquette in flour, shaking off any excess.

Dip the floured croquette into the beaten egg, then roll it in the breadcrumbs to coat fully. Transfer each coated croquette to a baking sheet or plate. Repeat with remaining croquettes.

Pour neutral oil into a medium skillet to a depth of 1/2 inch. Heat the oil over medium-high heat until it reaches about 350°F or is shimmering and hot.

Fry the croquettes in batches, without crowding the pan, turning occasionally so they brown evenly. Fry until golden brown on all sides.

Transfer fried croquettes to a paper-towel-lined plate to drain any excess oil. Garnish with additional chopped parsley and serve warm.