Bring 5 cups water to boil in large saucepan. Add chicken; cover and simmer until cooked through, about 13 minutes. Using tongs, transfer chicken to platter; reserve cooking liquid in saucepan. Cool chicken slightly. Remove chicken from bones; discard skin and bones. Shred chicken.
Set aside.

Add hominy, stewed tomatoes, oregano, jalapeño and hot pepper sauce to reserved cooking liquid. Bring to boil. Reduce heat; simmer until slightly thickened, stirring often, about 1 hour.

Stir in shredded chicken. Season with salt, pepper and additional hot pepper sauce, if desired. (Can be prepared 1 day ahead. Cover; chill. Bring to simmer before serving.)

Divide shredded lettuce, sliced radishes and sliced green onions equally among soup bowls.
Ladle posole into each bowl. Top posole with grated Monterey Jack cheese and serve. 

Serves 4 to 6.