Spray a slow cooker with nonstick cooking spray. Place chicken breasts in the slow cooker.
Heat olive oil in a skillet until shimmering. Add onion; cook and stir for 5 minutes. Add mushrooms and cook until onion is translucent and mushrooms are soft, 2 to 4 minutes. Add garlic and cook for 1 minute more.
Transfer the onion mixture to the slow cooker. Add chicken broth, taco seasoning, salt, and pepper. Cook on High for 1 hour 30 minutes.
Meanwhile, set oven rack about 6 inches from the heat source and preheat the oven's broiler. Line a baking sheet with aluminum foil. Place chile peppers cut-side down on baking sheet.
Broil the chile peppers until skin is heavily blistered and turning black, about 4 minutes per side.
Transfer the chile peppers to a bowl and cover tightly with plastic wrap. Allow to steam as they cool, about 15 minutes. Remove the skins and chop the flesh into 1-inch pieces.
Add Hatch chiles and heavy whipping cream to the slow cooker and cook for 30 minutes more.