In a medium bowl, stir together the garlic and lemon juice; set aside for 10 minutes. Meanwhile, in a small skillet over medium, heat the butter until bubbling. Add the Aleppo pepper and cook, swirling, until the butter is bright red and fragrant, about 1 minute. Remove from the heat and set aside.
To the garlic-lemon mixture, stir in the yogurt and ½ teaspoon salt. Divide among 4 serving bowls, then set aside at room temperature. In a small bowl, toss together the cilantro, mint and dill; set aside.
Fill a 10-inch skillet with at least 1 inch of water and bring to a simmer over medium-high. Line a large plate with paper towels. Crack each egg into a small bowl or cup. One at a time, carefully tip the eggs into the water near the edge of the pan. Once all eggs have been added, cover the pan and turn off the heat. Let stand until the whites are set but the yolks are still soft, 3 to 5 minutes.
Using a slotted spoon, gently transfer each egg to the prepared plate to briefly drain, then place 1 egg in each bowl, setting it on the yogurt. Drizzle each serving with 1 tablespoon of the Aleppo butter and top with the herb mixture, dividing it evenly. Sprinkle with sea salt, if using.
