Place the chicken in a large pot and cover with the cold water. Bring to a boil over medium-high heat.
Reduce the heat to low and skim the foam off the top with a large spoon. Discard the foam.
Add the onion, carrots, celery, garlic, salt, peppercorns, oregano, bay leaves and parsley.
Cover and simmer gently, with just a few small bubbles breaking the surface, until the chicken reaches 165°F (74°C) in the thickest part. Boneless breasts take 25 to 30 minutes and bone-in breasts take 40 to 45 minutes.
Turn off the heat and let the chicken rest in the broth for 10 to 15 minutes.
Transfer the chicken to a bowl and shred it with two forks while it's still warm. Discard the bones for bone-in chicken.
Pour 1 cup of the broth over the shredded chicken and toss.
Strain the remaining broth through a fine-mesh strainer and discard the solids. Taste and add more salt as needed. Cool before storing.