Bring chicken stock to boil. Add chicken breasts and simmer until cooked through—about 15 minutes. Transfer chicken to a plate. Reserve poaching liquid. (If you don’t have 3 cups remaining, add more chicken stock.) Shred the chicken and set aside.
Remove and discard husks from the tomatillos. Rinse and coarsely chop. Put tomatillos in a large bowl and add onion, cilantro, garlic and jalapeño. In a food processor, process half of this mixture with ½ cup water until chunky. Repeat with remaining half of mixture and another ½ cup water.
Place the tomatillo mixture in a large, deep-sided pot. Add the reserved chicken poaching liquid, shredded chicken, hominy and oregano. Simmer and cook until all ingredients are just heated through—2-3 minutes. Taste soup and season with salt as needed.
To serve ladle soup in 6-8 soup bowls and add garnishes.