Bring broth and ginger to a boil, covered, over high heat. Meanwhile, tear turkey into large shreds; set aside.
Add noodles, mushrooms, and broccolini to broth (if using dried noodles, cook noodles first for 2 minutes, then add vegetables). Cook noodles until almost tender, about 2 minutes for fresh and 4 to 5 minutes for dried.
Keep soup at a simmer and add turkey. Stir in hoisin and chili garlic sauces, then most of cilantro and green onions. Ladle into large, deep bowls and top with remaining cilantro and green onions.
Find fresh noodles in the refrigerated section of most grocery stores and Asian markets, and dried noodles in your grocery store's Asian-foods aisle.