In a large 12'' saucepan or skillet (make sure it has a tight-fitting lid), heat the olive oil over medium-high heat. Add the onion, green bell pepper, and diced jalapeño (keep the seeds for more spice, if desired). Sautée until the onions are translucent.
Add the garlic and fry for 1 more minute. Mix in the green chiles until combined, then pour the diced green tomatoes.
Cook the mixture until the tomatoes begin to soften and break apart. If needed, use a potato masher or immersion blender to break up the tomatoes for a chunky, sauce-like consistency.
Add the ground cumin, sugar, salt, pepper, and lime juice to the shakshuka base. If the sauce is too thin, let it simmer, uncovered, for 5-10 minutes.
Make 6 wells into the salsa verde sauce with a large spoon. Crack an egg into each well, then cover the skillet and adjust the heat to low. Let the eggs cook for 7 to 10 minutes, until the whites are set. The yolks should be cooked just under your liking, as the residual heat will continue to cook the eggs as it sits.
Remove the skillet from the heat and garnish the shakshuka with feta, cilantro, and lime wedges. Serve warm with tortilla chips or toast.