Place potatoes in medium saucepan with enough water to cover by one inch. Bring to a boil, then gently boil potatoes, until tender when pierced with knife, 15 to 20 minutes. Drain potatoes and let cool. Peel and cut into 1/4-inch-thick slices.
Meanwhile, in large bowl, whisk eggs until frothy. Add cheese, parsley, salt and pepper, and whisk until blended.
Preheat oven to 350°F In large ovenproof skillet, heat oil over medium heat. Add onion and cook, stirring often, until lightly golden, 10 to 12 minutes. Add garlic and cook, stirring often, 1 minute. Add butter and reduce heat to medium-low. Add potatoes and mix. Slowly pour eggs into pan; do not stir. Cook, undisturbed, 2 minutes.
Transfer skillet to oven and bake until eggs are cooked through, 15 to 18 minutes. Remove from oven and let sit 2 minutes, then loosen with spatula and slide onto serving plate. Let cool 5 minutes before slicing.