In a medium size bowl, mash the avocados with lime juice, salt, cumin, and cayenne. Stir in the jalapeno, tomatoes, cilantro, and garlic. Cover and set aside.
For the jalapeno mayo:
Mix all the ingredients together in a small bowl, then set aside.
For the burgers:
In a medium skillet, heat up ½ tablespoon extra virgin olive oil over medium heat. When the oil is hot, add the onions and cook for 2 minutes, stirring occasionally. Add the garlic, then cook for another 3 minutes, until the onions are soft and translucent. Remove from the heat, then set aside to cool (about 20 minutes).
In a medium mixing bowl, mix together the ground beef, salt, pepper, and cooled onion mixture with your hands. Form four round patties (make the patties slightly larger than your buns), then set aside on wax paper.
Heat a grill pan or a skillet over medium heat. If you're using a skillet, add 1 tablespoon extra virgin olive oil. Cook the patties on one side for 5-6 minutes for medium-rare. Flip the patties, then top each patty with a slice of pepper jack cheese and cook for another 5-6 minutes (or until desired doneness).
Place the buns, cut side down, onto a clean skillet or grill pan over medium heat. Toast for 30 seconds, until slightly golden brown.
Spread the jalapeno mayo on each toasted bun. Top the bottom buns with a patty, then add one or two spoonfuls of guacamole on top. Top with a handful of spring mix and the top bun.