1. Preheat broiler with oven rack 6 inches from heat. Line 2 rimmed baking sheets with aluminum foil. Spread tomatillos, onion, garlic, and jalapeño on 1 prepared baking sheet. Broil until blistered and lightly charred, 4 to 5 minutes per side. Remove from oven and let cool for 20 minutes. Reduce heat to 425°F.
2. Process charred vegetables, lime juice, and 1/4 cup of the cilantro in a food processor until smooth. Transfer to a bowl and stir in 1/2 teaspoon of the salt. Chill until ready to use.
3. Cut each avocado in half; remove and discard pits. (To keep avocados level, slice about 1/4 inch from the back of avocado halves so they sit flat.) Enlarge the hole by scooping 1/8 to 1/4 inch of flesh from around the hole. (Save the scooped avocado flesh for another use.) Sprinkle avocados with pepper and remaining 1/4 teaspoon salt; place on second prepared baking sheet.
4. Bake at 425°F until no residual moisture remains in the hole, about 8 minutes. Remove from oven. Working with 1 egg at a time, break eggs into a bowl and, using the eggshell, scoop 1 yolk into each hot avocado half. Add enough egg whites to fill hole completely. (Discard remaining egg whites.) Cover avocados with foil and return to oven. Bake until egg whites are set and yolks are runny, about 5 minutes.
5. Meanwhile, heat a medium skillet over medium-high heat and add oil and diced ham; cook until ham is browned and edges are crispy, about 4 minutes. Sprinkle avocado halves with the crispy ham and remaining chopped cilantro. Serve with salsa verde.