Place the peeled, chopped potatoes in a large pot and cover with water. Season the water with 1 tablespoon salt and bring to a boil over high heat. Reduce the heat and continue to simmer for 15 minutes or until the potatoes can easily be pierced with a fork. Drain the potatoes and mash until smooth. Stir in butter and season with salt and pepper.
Heat 1 teaspoon olive oil in saute pan over medium heat. Add chopped mushrooms and saute until browned, about 5 minutes. Season with salt and pepper. Remove from heat and add to the mashed potatoes. Stir to combine and let cool completely.
Whisk one egg and mix with the potato and mushrooms along with 1 tablespoon flour. Shape the potato-mushroom mixture into croquettes. Whisk the remaining 2 eggs in one bowl and pour the panko in another bowl. Dip the croquettes in the eggs and coat in the panko, shaking off any excess. Transfer the croquette to a lined baking sheet and continue with the remaining potato mushroom mixture. Cover and chill for 30 minutes.
Preheat oil to 375 degrees F.
Remove croquettes from the refrigerator. Carefully drop croquettes into the hot oil and fry until golden brown. Remove with a slotted spoon and transfer to a paper towel-lined plate. Repeat with remaining croquettes. Sprinkle grated parmesan on top if desired. Serve immediately.