Sear the Chicken: Prepare your slow cooker by lightly greasing the insert. In a spacious skillet, warm 1 tablespoon plus 1 teaspoon of olive oil over a medium-high heat setting. Generously season the chicken pieces on all surfaces with the kosher salt and ground black pepper. When the oil is shimmering, carefully place the chicken in the hot skillet, ensuring not to overcrowd it (work in batches if necessary). Sear for 4-5 minutes on the initial side until a deep golden crust forms, avoiding unnecessary movement. Turn the chicken and brown the other side for approximately 2-3 minutes. Relocate the seared chicken to the prepared slow cooker.
Sauté Aromatics: Lower the heat under the skillet to medium. If there are excessive browned bits, carefully use a paper towel to wipe the skillet clean. Introduce the remaining 2.5 teaspoons of olive oil. Once heated, add the chopped large onion and sauté for about 3-4 minutes, until it starts to become tender. Stir in the minced garlic and 1 tablespoon plus 1 teaspoon of balsamic vinegar. Continue cooking for another 60-90 seconds, allowing the garlic to release its aroma and the vinegar to concentrate slightly. Scrape these aromatics and any flavorful remnants from the pan into the slow cooker.
Combine and Slow Cook: Distribute the can of crushed tomatoes, the can of diced tomatoes with green chilies (undrained), the chopped large green bell pepper, the sliced mushrooms, the Italian seasoning, and the bay leaf over the chicken in the slow cooker. Secure the lid.
For chicken breasts: Cook on LOW for 4.5 to 5.5 hours or on HIGH for 2 to 3 hours, until chicken is tender and registers an internal temperature of 165∘F (74∘C).
For chicken thighs: Cook on LOW for 5.5 to 6.5 hours or on HIGH for 3.5 to 4.5 hours, until thoroughly cooked to 165∘F (74∘C). Once cooked, gently take the chicken out of the slow cooker. You can serve the pieces intact, or if preferred, shred or chop the chicken. Remember to find and discard the bay leaf.
Finish Sauce & Serve: For an enriched sauce, if your schedule permits: Remove the lid from the slow cooker and switch the setting to HIGH. Allow the sauce to simmer and reduce for 45 to 60 minutes, which will intensify its flavors and create a richer consistency. Shortly before you're ready to serve, incorporate the remaining ¾ teaspoon of balsamic vinegar into the sauce. Sample the sauce and adjust with a pinch more salt or pepper if you find it necessary. If you've opted to shred or chop the chicken, reintroduce it to the thickened sauce, stirring to coat and rewarm. Alternatively, plate the whole chicken pieces and generously ladle the hunter's style sauce over them. Serve your rustic slow-simmered chicken hot, accompanied by your choice of suggested sides, and garnish with a sprinkle of fresh chopped parsley.