Prepare the aioli: mix all ingredients together and leave in refrigerator for 30 minutes while you prepare the burgers.
For the burgers, sauté the shallots in a tablespoon of avocado oil until soft and translucent, about 10 minutes over medium-high heat. Add the garlic and cook until fragrant, about one more minute. Set aside to cool.
Combine all meats and egg and add cooked shallot and garlic mixture. Mix to combine and then divide into 4 patties. Shape patties into rounds, flattening them down and press a large dimple into the middle of the patty (this will prevent it from bowing up in the center when cooked).
Season with salt and pepper just before cooking (helps prevent texture changes from the salt exposure).
Grill or cook in a cast iron skillet over high heat, about 3-5 minutes per side (160ºF internal), or to your liking.
Serve on top of a small bed of arugula with a aioli, 1/4 avocado, and dill pickle, if desired.