Bring a large pot of heavily salted water to a boil and fill a large bowl with ice water. Add spinach and Swiss chard and blanch for 2 minutes, or until bright green, then transfer to ice water. Using tongs, transfer blanched greens to a high speed blender or bowl of a food processor. Blend until completely pureed, about 30 seconds. Set puree aside.
Heat oil in a 10-12 inch nonstick or cast iron skillet over medium heat. Add onions and saute until soft and translucent, about 7 minutes. Add garlic and serrano pepper and saute until fragrant, about 2 minutes. Add cumin, caraway seeds, and Aleppo pepper or harissa powder and cook until fragrant, about 1 minute. Season to taste with kosher salt and freshly groundblack pepper.
Add pureed greens, cilantro, parsley, and 1 cup of water to the skillet. Simmer until water has reduced by half, 3-5 minutes. Stir in heavy cream and cook until simmering, 2 minutes. Season to taste with lemon juice, salt, and pepper.
Using the back of a spoon, make wells in the mixture for the eggs. Reduce heat to medium low and gently add an egg into each well and cook for 7-9 minutes, depending on the desired doneness of the egg yolk.
Top with crumbled feta and red chili flakes. Serve immediately with crusty bread or pita.