Heat oil in a skillet. Add the ground meat and cook, breaking it down with a wooden spoon for 6-7 minutes, until browned. Add onion and garlic. Add carrots (optional).Add salt, black pepper, dried mint, summer savory, thyme (optional), tomato sauce and tomato paste. Cook for 2-3 minutes, stirring frequently.
Slowly add the milk, whisking to incorporate. Add salt and nutmeg. Cook for 3 minutes, until thickened. Let the mixture cool for 5 minutes. Add in the egg yolks (with feta). Let the sauce stand at room temperature, while the potatoes and meat are cooking.
Pour the sauce over the cooked potatoes. Top with cheese. Bake for 15 minutes, until golden. You can broil for 1-2 minutes, if the top is not getting to the desired golden color.