Step 1Heat a skillet over medium-high heat until hot. Add 2 tablespoons olive oil, then the onion and cook, stirring frequently, until the onion is softened, 5 to 7 minutes. Remove from heat and set aside to cool.
Step 2Place the romaine, fenugreek, green onions, dill sprigs and spinach and kale leaves in the bowl of a food processor. Pulse to finely chop.
Step 3Heat a large skillet over low heat. Add 2 tablespoons oil, along with the finely chopped greens, ½ teaspoon salt and several grinds of pepper, or to taste. Cook, stirring frequently, until the greens are nice and soft (the greens will have released their liquid and reduced by about half), about 15 minutes. Remove from heat, stir in the cooked onions along with a pinch of Aleppo pepper, and set aside to cool completely. Stir in the yogurt and barberries. (The mixture can be made ahead to this point and refrigerated up to 1 day in advance before continuing.)
Step 4Heat the oven to 350 degrees. Remove the kuku mixture from the refrigerator and spread it out in an oiled, oven-proof skillet, making four indentations for the eggs. Crack the eggs on top of the kuku mixture and place in the oven.
Step 5Bake the kuku until the eggs are sunny-side up, 15 to 20 minutes. Remove, sprinkle with feta cheese and another pinch of Aleppo pepper, and garnish with the four dill sprigs.
Step 6Serve immediately alongside grilled ciabatta or flatbread.