Chef Johns Chicken Tinga

Name: Chef John's Chicken Tinga
Ingredients: 3 pounds skinless, boneless chicken thighs; 4 cups chicken stock; 1 cup water; 1 large yellow onion, halved; 3 peeled whole garlic cloves; 2 teaspoons kosher salt, plus more to taste; 0.5 teaspoon dried oregano; 2 tablespoons olive oil; 1 large onion, diced; 1 (7 ounce) can chipotle peppers in adobo sauce; 1 (28 ounce) can peeled plum tomatoes (preferably San Marzano); 3 tablespoons chopped fresh cilantro, or more to taste; 2 tablespoons crumbled cotija cheese, or to taste