Preheat the oven to 375ºF.
Heat the oil in a 12-inch cast iron pan or ovenproof skillet over medium heat. Add the tomato paste, onion, pepper, tomatillos, and chard; sauté for 3 to 4 minutes. Add the garlic and cook until fragrant, about 30 seconds. Add the salt and simmer for about 10 minutes.
Make 4 wells in the sauce and gently break one egg into each. Transfer to the oven and bake until the whites are cooked and yolks are set, 10 to 12 minutes.
Allow to cool for 3 to 5 minutes, garnish with cilantro, and carefully spoon onto plates.
