Heat 1 tablespoon oil in a large skillet over medium heat.
Add the mushrooms and ½ teaspoon salt. Saute, stirring occasionally, for 5 minutes. Stir in the minced garlic and saute for another 5 minutes, until the mushrooms are deeply browned. If the mushrooms start to release water, continue cooking them until most of the water has evaporated, about 3-4. (Depending on the type of mushroom, they might not release much or any water.)
As the mushrooms are cooking, peel and chop the onions and core and chop the peppers. Once the mushrooms are ready, stir in the onions and peppers to the skillet, plus 1 teaspoon salt and freshly ground pepper.
Transfer the veggie mixture to your slow cooker insert. Stir in the drained olives and dried herbs and pour in the broth. Nestle the chicken into the stew - it won't be totally submerged and that's okay. Season the chicken with salt and pepper.
Set the slow cooker to low for 4 hours. When it's done, use two forks to shred the chicken into bite-sized chunks (if your slow cooker has a nonstick insert, you might want to pour the stew into a bowl to do this so the forks don't scrape the nonstick coating), or move them to a cutting board and cut into bite-sized chunks.
Add the halved cherry tomatoes to the veggies and stir everything together. Stir the chicken back in and taste for seasoning, adjusting as necessary. Serve warm, refrigerate for up to 5 days, or freeze for up to 3 months.