In a slow cooker, mix together 1 cup white onion slices, 1 tablespoon chopped fresh garlic, ½ red jalapeño pepper (sliced; or 1-2 Thai chiles, halved), 3 slices fresh galangal (or fresh ginger, each slice ¼-inch long), 1 stalk lemongrass (pounded with side of knife, cut into 2-inch-long pieces), 10 lime leaves (torn, optional), 1 tablespoon red Thai curry paste, and 4 cups chicken broth. Stir well to blend all elements.
Cover the slow cooker with its lid. Set it to cook on HIGH for 3 hours or on LOW for 6 hours, stirring from time to time.
After the cooking time is done, take out the solid pieces with a slotted spoon, leaving only the liquid in the cooker.
Pour in 2 13.5-ounce cans full-fat coconut milk (or coconut cream, at room temperature), add 1 pound chicken breasts (cut into bite-sized pieces), and 8 ounces white mushroom caps (sliced). Stir to combine everything.
Replace the lid on the slow cooker. Continue cooking on HIGH for 1 hour or on LOW for 2 hours, until the chicken breast pieces are fully cooked through.
Once the chicken is done, stir in 2 tablespoons brown sugar (or coconut sugar), 2 tablespoons fish sauce, and 2 tablespoons fresh lime juice. Mix well, then taste and adjust the amounts of brown sugar, fish sauce, and lime juice to your preference.
When you are happy with the soup's flavor, spoon it into serving bowls. Top each bowl with thinly sliced green onions and chopped fresh cilantro, and serve right away.