Heat oil in a large saucepan over medium-high heat. Add fresh mushrooms, green onion, ginger, and salt. Cook, stirring occasionally, until mushrooms begin to brown, about 5 minutes. Add rice wine; cook, stirring and scraping browned bits from pan bottom, about 1 minute.
Add dried mushrooms, dried kombu, date sugar, and vegetable broth. Simmer over medium-low heat until reduced by 3 cups, about 40 minutes. Strain and discard solids.
Add goji berries and 1/4 cup sliced ginger to broth. Keep warm, covered, or chill, covered, up to 3 days. (Or freeze in an airtight container up to 3 months.)
Serve broth tableside over a heat source to maintain a rigorous simmer. Cook desired ingredients in broth; serve with condiments.