Cut up the garlic, carrots, and onions. You'll be pureeing them later, so they don't have to be chopped finely. Place the vegetables in a large pot or Dutch Oven over medium-high heat, and sauté until they begin to soften.
In a large pan on medium heat, melt a tablespoon of oil or butter. Sear the chicken thighs on both sides, until they reach a golden brown.
Add the chipotle peppers and bay leaf to the softened vegetables. Let them continue to cook together until the chicken thigh have finished cooking in the hot skillet.
Add the chicken thighs to the pot with the peppers and vegetables. Pour in hot water, bouillon cubes, and all of the spices and seasonings.
Keep the pot on medium high heat, and wait until it comes up to a boil. Let the pot boil for 30 minutes.
Remove the braised chicken from the pot, strain out the vegetables, and place the chicken back in the both with the strained broth.
Put the vegetables in a blender and blend until smooth. Add this vegetable puree back into the pot with the broth and the chicken.
Let the pot simmer over medium heat for 3 hours. Remove from heat.
Shred chicken while it's still in the pot, using two forks. Once the chicken is shredded, strain it from the sauce and remove the fatty pieces and bones.
Heat a large skillet, and prep the ingredients for the tacos nearby. This includes the birria sauce, the shredded chicken, the corn tortillas, chopped onions, chopped cilantro, and cheese.
Add a tablespoon of olive oil or vegetable oil to the skillet over medium-high heat. Dip a tortilla into the sauce, making sure both sides are fully coating. Place the tortilla in the hot pan. Fill one half of the tortilla with chicken and the other half with cheese. Sprinkle on a few onions.
Use a spatula to flip half of the tortilla on top of the other half, making it a taco shape. Use the spatula to press the taco down, searing it well and melting the cheese.
Continue making tacos until the chicken is used up, and then serve the the remaining dipping sauce sprinkled with cilantro.