How to Make It
Heat oil in a large Dutch oven over medium-high. Sprinkle chicken with 1 teaspoon of the salt. Cook until browned, 4 minutes per side. Transfer to a plate; reserve drippings in pan. Add onion, poblanos, and tomatillos to hot drippings. Cook, stirring, until softened, 3 minutes. Stir in garlic, cumin, and thyme; cook 1 minute. Stir in broth and remaining 1 teaspoon salt.
Add chicken; cover and cook over medium-low until tender, about 40 minutes. Remove from heat. Transfer chicken to a cutting board; cool 5 minutes. Remove and discard bones. Chop chicken; return to Dutch oven.
To serve now: Stir in hominy. Top soup as desired.
To freeze for later: Cool completely, 30 minutes. Transfer to ziplock plastic freezer bags. Freeze up to 3 months.
To reheat and serve: Thaw bags of soup slightly in warm water. Peel off and discard bags as you place soup in a saucepan; add hominy. Cover; cook over medium-low 1 hour. Top as desired.