Spray a 6 quart Slow Cooker lightly with non-stick spray. Add the squash, peppers, onion, garlic, hominy, chilies, and black beans to the cooker and stir to combine.
Combine the pureed beans, chicken stock, and red chili sauce in a bowl. Add the masa harina and stir or whisk until the mixture is smooth.
Add the chili powder, cumin, and oregano to the liquid ingredients and stir once more. Pour the liquids into the cooker and stir to combine everything well.
Cut the chicken into bite size pieces. Lay the chicken pieces on top of the vegetable mixture. Cover and cook on Low for 6 - 8 hours (the heat on every slow cooker is different so adjust the time according to how yours cooks).
At the 6/8 hour point, stir the chicken into the other ingredients. Give things a taste and add more of any seasoning you think it needs.
Tilt the lid slightly and turn the heat setting to High. Allow the chili to cook until it is the thickness that you like.
While the chili is cooking on High, mix the sour cream, salsa, and cilantro together. Set aside.
When you are ready to serve, ladle into bowls. Pass the sour cream mixture, lime, and cheese at the table.
Place the oats and garlic in the bowl of a food processor. Pulse until they are coarsely chopped.
Add half of the beans, the egg and the cumin to the bowl. Process until it is a fairly smooth paste, scraping down the bowl as needed.
Put the bean paste into a medium bowl. Add the chopped onion, cilantro, and the remaining whole beans. Grate half of the cheese finely and add to the bowl. Stir everything until the mixture is well mixed.
Scoop the mixture out into 4 equal portions. Gently form into pattys. Spray a plate lightly with the cooking spray. Place the burgers on the plate cover with plastic wrap and chill for at least 1 hour (more won't hurt).
Heat a large non-stick skillet over medium- high heat. Add the olive oil and swirl to coat the pan. Place the burgers in the pan and cook without moving for 3 or 4 minutes. They should be nicely browned.
Flip the burgers and cook for another 3 - 4 minutes. Thinly slice the remaining cheese and place a slice on each of the burgers. Cover for 2 minutes or until the cheese is melted.
Put the bottom of the bun on a plate. Top with the shredded lettuce and drizzle with the salsa. Place the burger on top and add the tomato and avocado and serve.