Cook tomatoes and tomatillos in water over medium heat for 10 minutes until softened. Allow to cool slightly and place in a food processor and pulse until smooth.
Heat the olive oil in a deep pan over medium heat. Add the onion slivers and cook until softened (3 minutes). Stir in the chopped garlic and cook for an additional 1 minute.
Add the sauce to the pan. Then add the oregano, salt, pepper, and adobo sauce. Simmer for 5 minutes and taste. Add additional adobo sauce per your taste and cook an additional 5 minutes.
Stir in the shredded chicken to combine. Stir occasionally for 10 minutes and taste. Garnish with some chopped cilantro.
If you prefer a hotter and more smoky flavor add one chipotle chile from the can of adobo sauce to the food processor. Boil chicken breasts in water for 15 minutes, cool, and shred with two forks. You can also use pre-cooked rotisserie chicken for shredding.