Heat a large saucepan over medium, add the butter and olive oil and brown the lamb. Toss in the garlic and oregano, give it a few shakes and stir for a minute then pour in the water. Season with the salt and pepper, adding more if you wish.
Lower heat to a gentle simmer, cover with a lid and cook for 1 hour; stirring once or twice during cooking.
Stir the rice into the liquid, put the lid back on and cook for another 5 minutes. Turn off the heat.
Decant the lamb and juices into a deep baking dish (at least 8cm deep) with lid. Set aside.
In a separate small pan melt the butter over medium heat, add in the flour and whisk for 1-2 minutes as it foams and thickens. Turn off the heat and set aside.
Whisk together the yoghurt and eggs in a mixing bowl and season well with salt and black pepper. Whisk in the roux until emulsified, then pour it over the lamb evenly. Scatter over a little more oregano and bake in the oven for 40-50 minutes, or until golden.
Once done, let the tavë kosi rest for 10 minutes before serving.