In a 10-inch skillet, heat the oil over medium heat until shimmering. Add the onion and garlic, and season with salt and pepper. Cook until the onion begins to soften, which usually takes about 2-3 minutes. Then, add the parsley and continue cooking for an additional 2 minutes, or until the parsley starts to soften and turns bright green.
Add the tomatoes, Parmesan, gochujang, and brown sugar (if using), and stir everything together thoroughly. Season with salt and pepper, then reduce the heat to medium-low and let it simmer for about 15 minutes, stirring occasionally. Taste the mixture and adjust the seasonings if necessary. If you're using greens, add them to the skillet and cook until they are just barely wilted. More delicate greens like spinach will only take about 30 seconds to 1 minute, while tougher greens like kale will require 3 to 4 minutes.
Make small indentations in the sauce for the eggs, one for each egg, and crack them into the pan. Adjust the heat to medium, cover the pan, and cook until the whites are set and the yolks are still soft, which typically takes about 5 to 7 minutes (or longer if you prefer firmer eggs). While the eggs are cooking, toast your bread. Once the eggs reach your desired level of doneness, remove the pan from the heat and serve the dish warm, sprinkled with additional Parmesan and parsley. Enjoy it with crunchy toast for dipping.
