Make aioli: Combine all ingredients for aioli in small bowl. Cover and refrigerate until ready to use.
Prepare toppings: Toast buns, and slice veggie toppings, set aside.
Make patties: Process mushrooms, poblano, onion, and garlic in food processor until finely chopped. Scrape into medium mixing bowl.
Add ground turkey, Greek yogurt, 2 Tablespoons avocado oil, salt, pepper, chili powder, cumin, coriander, paprika, and garlic powder. Combine until evenly mixed without over-working meat.
Pat into 4 large or 6 medium patties.
Heat remaining 1 T avocado oil on medium in large cast-iron skillet. Add patties to pan and cook 4 minutes. Flip and cook 2 more minutes.
Carefully add ⅓ C water to bottom of pan, cover lightly with tin-foil or lid and cook 4 more minutes. Place one slice of cheese on top of each patty an re-cover. Cook about 2 more minutes - until internal temp of 165℉ is reached. Add small amount of water as needed to keep from drying out.
To serve: Spread buns with aioli, add patty, top with toppings.