Place vegetable stock, water, and lobster base in a large stockpot; bring to a boil. Stir in potatoes; boil until tender, about 15 minutes. Drain potatoes, reserving stock.
Melt butter in the stockpot over medium heat. Add onions and tarragon; cook and stir until onions are translucent, about 5 minutes. Add arrowroot; stir until thoroughly combined. Pour reserved stock back into the stockpot. Stir in half-and-half and bring bisque to a boil. Return potatoes to the stockpot; lower heat and simmer until heated through, about 5 minutes.