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Combine shiitake and wood ear mushrooms in a small bowl; add warm water to cover. Let soak until plump, about 1 hour. Drain, reserving soaking liquid. Set mushrooms and soaking liquid aside separately.
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Place chicken in a large bowl; pour wine over chicken. Place scallions inside cavity of chicken. Set aside.
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Combine onions and ginger in a large pot. Place chicken on top; add wine from bowl, bay leaves, mushrooms, and mushroom soaking liquid. Add enough water to cover chicken. Bring to a boil over high. Using a mesh skimmer, skim foam on surface of broth in pot. Reduce heat to medium-low, and simmer, uncovered, 2 hours.
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Add jujube dates, goji berries, astragalus root, and angelica root to pot, and stir to submerge in liquid. Cook broth mixture over medium-low, uncovered and undisturbed, until chicken is very tender and broth has reduced by about 2 cups, 45 minutes to 1 hour. Remove from heat.
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Remove chicken from pot, and transfer to a cutting board. Let rest until cool enough to handle, about 20 minutes. Remove and discard skin and bones; reserve all meat. Pour remaining broth mixture in pot through a fine wire-mesh strainer into a large heatproof pot, and discard solids. Stir in sea salt. Set bone broth aside.