Heat 2 Tbsp olive oil in a large skillet, and add the onions and garlic. Sauté until soft and translucent. Add the mushrooms and sauté until softened, then add the kale, stir a few times, and cover the pan. Cook down until the kale is softened and has reached a deep green color. Remove pan from heat, and set aside to cool.
When the mushroom/kale mixture has cooled, combine it with the turkey meat, oregano, chimicurri, and salt, and stir vigorously to combine. You can also use your hands if it's easier.
Heat remaining 2 Tbsp olive oil in a large skillet.
Portion the burgers into small patties (mine were about 3-4 Tbsp worth of the turkey mixture) and cook until crispy and browned on the bottom, about 2-4 minutes. Flip, and repeat on the other side. They are done when the burgers feel firm when pressed and the juice that runs out is clear.
To assemble: Coat the buns with mayo and chimichurri sauce, and top with the burger and avocado (if desired). For a paleo option, serve burger topped with mayo and chimichurri, or just additional chimichurri on top of the bed of slaw and garnish with avocado slices.