Bring 4 cups chicken stock to a boil in a large, heavy saucepan (with a lid). Add chicken breasts, cover, and simmer until cooked through, for about 14 minutes longer. Using tongs or a slotted spoon, transfer chicken to plate.
Reserve poaching liquid. You should have 3 cups liquid; if not, add enough stock to equal this amount.
Shred the chicken and set aside. (Soup can be prepared to this point one day ahead. Cover poaching liquid and chicken separately and refrigerate.)
Remove and discard husks from tomatillos; rinse and coarsely chop. Put chopped tomatillos in a large bowl; add onion, cilantro, garlic, and jalapeño.
In a food processor or blender, process half of this mixture with ½ cup water until chunky; repeat with remaining half 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. Bring mixture to a simmer and cook just until all ingredients are heated through, for 2 to 3 minutes.
Taste soup and season with salt, as needed.
To serve, ladle soup into 6 soup bowls. Arrange garnishes in small bowls and pass separately.