Heat a large skillet over medium heat. Add ground beef and ground pork and cook until brown and crumbly, 5 to 10 minutes. Drain and discard fat.
Add 1/4 cup olive oil, carrot, onion, celery, half the Italian parsley, and tomatoes. Mix to combine. Stir in pepper and season with paprika, salt, pepper, bay leaves, and cayenne pepper. Cook until vegetables start to soften, about 10 minutes.
Meanwhile, preheat the oven to 400 degrees F (200 degrees C).
Transfer meat mixture to a large 9×13 roasting pan.
Heat remaining 1/4 cup olive oil in a large skillet and cook potatoes until browned, about 10 minutes. Transfer to the baking pan and mix well with the meat mixture.
Bake in the preheated oven for 45 minutes. Remove baking dish from the oven and mix in remaining chopped parsley leaves.
Stir eggs, yogurt, flour, and baking soda together in a bowl until it turns into a spreadable mixture. Pour over the meat mixture in the baking dish.
Return baking dish to the oven and cook until the top is golden brown, about 15 more minutes.