Preheat the oven to 375°F/190°C
Heat a large saucepan over medium-high and melt the butter and add olive oil.
Toss the lamb into the saucepan and cook for about ten minutes or until the meat is browned. Add garlic, oregano, stir for a minute then pour in the water. Adjust with salt and pepper s your taste.
Lower heat to low, after you bring it to boiling, cover with a lid. Cook until the rice is tender or for 20-30 minutes, stir during cooking when necessary.
Stir the rice into the liquid, put the lid back on and cook for another 5 minutes. Turn off the heat.
Then transfer lamb mixture to a casserole dish or a deep baking dish (at least 8cm deep) with a lid and set aside.
You can use a separate small pan over medium heat and melt the remaining 2 tbsp of butter. Add in the flour and whisk for 1-2 minutes then set aside and turn the heat off.
Use a mixing bowl to whisk together the eggs and the yogurt, add some seasoning with black pepper and salt. Whisk well then pour the yogurt sauce evenly over the meat in the baking dish.
You can add more oregano if you'd like and bake in the oven for 40-50 minutes.
Let the tavë kosi cool for 10 minutes before serving.