Heat olive oil and onions in a large skillet over medium heat. Cook until soft, stirring often (about 5 minutes). Add garlic and cook another minute.
Pulse tomatoes and chipotles in food processor until almost smooth. Pour into onions and garlic.
Add cumin, oregano, salt, and chicken broth. Bring to a simmer and let cook for 15 minutes.
Add the shredded chicken to the sauce. Cook another 10 minutes until the chicken is heated through.
Warm tortillas per package directions. Add the chicken tinga mixture to the tortillas. Top with fresh cilantro, Pico de Gallo, and a dollop of sour cream.