Add chicken, onion, garlic, celery, carrot, the 2 tablespoons of salt, and pepper to a large stock pot with enough water to cover ingredients with an additional 3 inches of water. (You will need a total of 3 cups of broth for the mole sauce.)Bring ingredients to a boil over high heat for 10 minutes. Adjust to medium heat and continue to simmer for an additional 20 minutes until the chicken is completely cooked. In the meantime, add onions to a small bowl and toss with juice from limes, oregano, and salt. Set aside.
Next, remove the chicken from the broth and set aside to cool. Once cool enough to handle, strain the broth through a fine-mesh metal sieve and set aside for the mole sauce.
Add mole paste, peanut butter, chocolate and 3 cups of the chicken broth to a large heavy-bottom frying pan over medium heat. Cook, stirring with a wooden spatula, until mole paste, peanut butter, and chocolate have melted and turned into a thick sauce, for about 10 minutes. Taste and adjust for salt if necessary. Reduce heat to lowest setting and simmer while preparing the chicken.
Shred chicken in a bowl and season with additional salt, if necessary.
Heat ¾ cup of oil in a small skillet over medium-high heat until hot but not smoking.
Warm tortillas on a comal or wrapped in a clean dish towel in the microwave for 30 seconds.
Using tongs, prepare mole enchiladas one at a time by carefully dipping warm tortilla into hot oil to coat on each side, just long enough to soften, 10 to 15 seconds.Drain excess oil and then dip the tortilla in mole sauce and drain excess sauce before filling the tortilla with 2 tablespoons of shredded chicken and rolling the tortillas as tightly as if making taquitos. If rolling them is a challenge, you may simply fold in half with chicken inside. Serve mole enchiladas topped with additional mole sauce, sliced onion, and crumbled cheese alongside rice and refried beans.