Place the chicken breasts, pepper, garlic, 1 teaspoon salt, halved red onion, and oregano in a large pot; add enough water to cover the chicken.
Bring to a boil, then reduce heat, cover, and simmer until the chicken is very tender, about 20–30 minutes, adding a little water if needed.
Remove the pot from the heat and let the chicken cool to room temperature in the liquid or briefly on a plate; reserve the cooking liquid if desired as broth.
Remove the chicken from the liquid and shred it finely using two forks or your hands.
Heat the olive oil in a skillet over medium heat, add the shredded chicken, and cook, stirring, until heated through, about 3 minutes.
Pour in the tomato sauce and add the parsley; stir and cook until everything is heated and combined, about 2–3 minutes.
Taste and add remaining salt as needed, then serve warm or allow to cool to room temperature before using as a filling.
