Add the chicken broth, water, diced onion, minced garlic, and chicken tenders into your large pot. Bring it to a boil, then reduce to a simmer, cover, and let it cook for about 15 to 20 minutes. Once the chicken is fully cooked through, pull it out and shred it with two forks before returning it to the pot.
Stir the taco seasoning into the simmering broth so those spices have time to bloom and flavor the liquid. Add in the drained white beans, the can of green chiles, and the corn or hominy along with its liquid. Let this simmer gently for another 10 minutes so all the flavors come together nicely.
Lower the heat so the chili isn't boiling hard, then stir in the cream of chicken soup, sour cream, and shredded cheese. Whisking gently here prevents the sour cream from breaking. This is the step that turns a basic chili into something closer to a creamy, hearty stew.
In a separate bowl, combine the corn muffin mix with the egg and just enough liquid to form a thick, scoopable batter thicker than regular cornbread batter. You want it stiff enough to hold its shape when dropped into the pot instead of dissolving into the broth.
Spoon dollops of the thick batter directly onto the surface of the simmering chili, spacing them out so they have room to puff. Cover the pot with a tight-fitting lid and let them steam undisturbed for 12 to 15 minutes. Resist lifting the lid early, since escaping steam slows down the cooking.
Test doneness by inserting a toothpick into the center of a dumpling; it should come out clean with no wet batter. Once they're set, ladle the chili into bowls, making sure each one gets a few dumplings on top. Garnish with green onions and a spoonful of sour cream before serving warm.