Make the guacamole: Pulse all ingredients up to avocado in food processor until well combined. Transfer to bowl. Add avocado and mash with fork. Add lime juice. Set aside.
Make burgers: Combine all ingredients in bowl, mix with hand until well combined. Set in fridge.
Make plantain bread: Cut plantains on a bias, 1-inch thick. Set a large frying pan over high heat, fill with about 1/2 inch canola oil. Heat oil to 375 degrees. Fry plantains briefly until lightly golden. Remove, flatten with tostonera or a plate lined with parchment paper. Refry until crisp and golden brown. Set aside on paper towels to drain.
Cook burgers: Portion seasoned ground beef into four patties about 1/4 pound each. Fill pan with 2 tablespoons of canola oil, pan fry on high heat first then reduce (do not flatten with spatula) to medium heat. Cook to desired doneness.
Assemble burgers: place a plantain on a plate, top with burger, cheese, guacamole, and second plantain disc. Serve immediately.