In a small skillet, heat the olive oil over medium heat. Add the shallots and sweat them until soft and translucent. Transfer the shallots to a large bowl.
Add the ground turkey, cilantro, sriracha, salt and pepper. Mix with your hands until well combined.
Form the burgers into 4 equal patties and transfer to a sheet pan. Use the rounded part of a tablespoon to make an indentation in the center of each burger (to prevent the puffy dome from forming as they cook).
Refrigerate the burgers for 15-20 minutes to firm up.
Meanwhile, preheat the grill to a medium high heat (around 450°).
Make the Aioli
In a small bowl, combine the mayonnaise, sriracha, cilantro and lime zest. Stir and taste for seasoning. If you like it spicier, add more sriracha, for more tang, add more lime. Refrigerate until ready to use.
Grill the burgers
When grill is hot add the burgers and cook about 5 minutes per side or until they reach 165° internal temperature. During the last minute of cooking, place a slice of American Cheese on each burger and cook until it's just melted.
Transfer burgers to a cutting board to rest for a minute or two. Meanwhile, toast the buns on the grill until just lightly browned.
Assemble the burgers with sliced avocado on the bottom bun, followed by the burger, a tablespoon of sriracha aioli and extra cilantro leaves if desired.