Boil the Potatoes: Peel and dice the Idaho potatoes into evenly sized chunks. In a large pot, cover with salted water, bring to a boil, and cook for about 15 minutes. Drain and return to pot to evaporate moisture, then mash with butter, salt, and pepper to taste.

Sauté the Mushrooms: Heat olive oil in a skillet over medium heat. Add chopped mushrooms and sauté for 5–7 minutes until golden brown. Season with salt and pepper, then remove from heat and cool slightly before mixing with mashed potatoes.

Combine Ingredients: In the mixing bowl with mashed potatoes and mushrooms, add whisked egg and all-purpose flour. Mix until well combined. Adjust seasoning, then set aside to rest for a few minutes.

Shape the Croquettes: With slightly damp hands, form small portions into oval or cylindrical shapes. Place on a tray lined with parchment paper and chill in the refrigerator for about 30 minutes.

Prepare for Frying: Heat oil to 375°F (190°C). Set up a breading station with whisked eggs and panko breadcrumbs. Dip each croquette into egg, then coat with breadcrumbs.

Fry the Croquettes: Carefully place several croquettes in hot oil. Fry for 3–4 minutes per side until golden brown, turning with a slotted spoon. Drain excess oil on a paper towel.

Serve and Enjoy: Sprinkle grated Parmesan over croquettes if desired. Serve warm with dipping sauces like aioli or marinara.