Add the oil to a large non-stick skillet and heat to medium. Add the onions and cook, stirring occasionally, until onions begin to brown on the edges, about 15 minutes. Reduce the heat to medium-low and continue cooking, stirring occasionally, until onions are dark brown and caramelized, about 40 minutes. Add a small amount of water twice throughout the cooking process to deglaze the pan when the onions begin sticking.
Once the onions are almost fully caramelized, add the mushrooms. Continue cooking until onions are a deep brown color and mushrooms have leached mush of their liquid, have softened, and have a golden-brown sear on them.
While the onions are caramelizing, make the pesto sauce and prepare the burgers. Add the basil, walnuts, and sea salt to a food processor and pulse until chopped. Leave the food processor on while streaming the oil through the top. Taste for flavor and add more sea salt or oil as desired. Set aside until ready to use.
Add the ground meat to a mixing bowl along with the sea salt, onion powder, and garlic powder. Mix well with your hands. Form 4 to 6 burger patties (according to your size preference).
Add 1 to 2 tablespoons of avocado oil (or cooking oil of choice) to a large cast iron skillet and heat to medium-high. Once the skillet is fully hot, carefully place the burger patties on the surface. Cook until deep brown char marks appear, about 3 to 5 minutes. Carefully flip and continue cooking to desired doneness, another 2-3 minutes for medium-rare, 5 minutes for medium, and 7 to 8 minutes for well-done.
Toast your burger buns (if desired), then add sauteed kale (or green of choice), a burger patty, pesto sauce, and caramelized onions and mushrooms. Sink your teeth in, and enjoy.