Preheat the oven to 375 degrees F. Mist a baking sheet with cooking spray. Grate the carrot and 1/2 onion into a large bowl. Add the chicken, panko, 2 tablespoons hoisin sauce, the ginger, 3 teaspoons soy sauce and 1 teaspoon sambal oelek and mix until combined. Shape the chicken mixture into 4 patties and place on the prepared baking sheet. Bake until cooked through, about 20 minutes. Meanwhile, thinly slice the remaining 1/2 onion. Toss with the mushrooms, sesame oil, lime juice and the remaining 1 teaspoon soy sauce in a bowl. Mix the remaining 2 tablespoons hoisin sauce and 2 teaspoons sambal oelek with 1 tablespoon water in another bowl. Warm the hamburger buns in the oven. Serve the burgers on the buns with a drizzle of the hoisin-sambal sauce and some of the mushroom mixture.