Preheat the oven to 375°F (190°C).
Prepare the potatoes: Peel and slice the potatoes, then boil in salted water until they are tender. Drain and set aside.
Prepare the broccoli: Cut the broccoli into florets, steam, or blanch until slightly tender, then set aside.
Sauté the onion and mushrooms: Dice the onion and mushrooms. Heat olive oil in a skillet over medium heat and sauté the onion until softened. Add the mushrooms to the skillet and cook until they release their moisture and become golden brown.
Make the egg mixture: In a large bowl, whisk together the eggs and yogurt, then season with salt and black pepper.
Combine all ingredients: Add the cooked potatoes, broccoli, sautéed onion, and mushrooms to the egg mixture, stirring everything together.
Bake: Transfer the mixture into a greased casserole dish and bake for about 25-30 minutes or until the top is golden brown.
Prepare the sauce: In a small bowl, mix together the Greek yogurt, mayonnaise, and minced garlic clove.
Serve: Once the casserole is done, remove it from the oven and let it cool for a few minutes. Serve with the prepared sauce on top or on the side.