Peel and chop the potatoes into uniform pieces. In a pot, add the potatoes to salted boiling water and cook for about 15 minutes, or until fork-tender. Drain and cool.

In a skillet, heat olive oil over medium heat. Add onion and sauté for 3-4 minutes until translucent. Stir in mushrooms and garlic, cooking for an additional 5-7 minutes. Season with thyme, salt, and pepper.

In a mixing bowl, add sautéed mushroom mixture, grated cheese, and egg to cooled mashed potatoes. Mix until combined.

Shape the mixture into patties about 3 inches in diameter, ensuring they hold together.

Coat each potato cake in breadcrumbs on a plate.

In a skillet, heat olive oil over medium-high heat. Fry the cakes for about 4-5 minutes on each side until golden brown and crispy.

Transfer cakes to a plate lined with paper towels. Sprinkle with fresh parsley as a garnish.