1. Preheat the Oven: Preheat your oven to 375°F (190°C) and grease a 9×13 inch baking dish to prevent sticking.
2. Prepare the Potatoes: Take the 4 medium potatoes, cubed, and add them to a large pot of boiling water. Boil for about 8-10 minutes until they are just tender, then drain and set aside.
3. Sauté the Aromatics: In a skillet over medium heat, melt 2 tbsp of butter. Add in the diced onion (1 onion) and minced garlic (2 cloves). Sauté until fragrant and the onion becomes translucent, about 3-4 minutes.
4. Combine the Sauce: In a large mixing bowl, combine the can of cream of mushroom soup, ½ cup of sour cream, sautéed onion and garlic, 1 tsp thyme, and salt and pepper to taste. Mix well until fully combined.
5. Layer the Ingredients: Begin layering the casserole. Spread half of the drained potatoes evenly in the greased dish, followed by 2 cups of trimmed green beans, and half of the sauce mixture. Sprinkle ½ cup of shredded cheddar cheese on top of each layer. Repeat this layering process until all ingredients are used up, finishing with a layer of sauce and cheese.
6. Add the Finishing Touch: Top the casserole with ½ cup of grated Parmesan cheese for an extra cheesy crust.
7. Baking: Place the casserole in the preheated oven and bake for 25-30 minutes, or until the cheese is melted, bubbly, and lightly golden brown.
8. Garnish and Serve: Once out of the oven, let it cool slightly. Garnish with fresh parsley and serve warm.

Cooking until the cheese is perfectly melted and the edges are bubbly is key. This dish will not only fill your belly but also warm your heart.