Place chiles on a baking sheet, broil for 12 minutes, turning halfway through. Peppers should be starting to blacken and blister. Remove from oven, let cool enough to handle and remove stems (if heat isn't your thing, you can also remove the seeds).
Add chiles to food processor and pulse until finely chopped. Set aside.
Mix ground beef and all spices in a large bowl. Using your hands, mix the spices in with the ground beef.
Form the beef into four patties.
Heat a grill pan over medium heat. Once hot, place burgers on the grill pan and cook for ten minutes, flipping after five minutes.
While burgers are cooking, heat a skillet over medium heat and coat with olive oil. Crack eggs onto skillet, two at a time so they don't touch. Cook until yolks are firm, but runny on the inside, about five minutes.
Top each burger with a spoonful of the green chiles and a fried egg.