Set a large Dutch oven or soup pot over medium heat and warm the olive oil.
Add the onion, jalapeños, poblano, bell pepper, sweet pepper, and garlic. Stir often and cook for 5 to 6 minutes, until the onion turns soft and see-through.
Crumble in the ground turkey. Break it apart as it cooks, about 6 to 7 minutes, until no pink is left.
Sprinkle the 2 teaspoons red pepper flakes and 1 tablespoon all-purpose seasoning over the turkey and stir until it's coated.
Pour in the diced tomatoes, tomato sauce, tomato paste, refried beans, Great Northern beans, and corn. Stir until the refried beans melt into the sauce.
Add the BBQ sauce, beer, and chili seasoning packet and stir well.
Fold in the shredded chicken and the 1 teaspoon salt. Bring it to a gentle bubble, turn the heat to low, and simmer for about 15 minutes, stirring now and then.
Ladle into bowls and top each one with a spoonful of Greek yogurt and a few jalapeño slices, if you like.