In a large Dutch oven, heat oil over medium-high heat. Add onion and fennel; cook, stirring occasionally, until tender, about 5 minutes. Add garlic; cook until fragrant, about 1 minute. Add broth, potatoes, wine, 1 cup water, salt, and pepper; bring to a boil. Reduce heat to medium; cover and simmer until potatoes are fork-tender, about 15 minutes. Reserve ¼ cup asparagus tips for garnish. Add remaining asparagus; cook until tender, about 4 minutes. Remove soup from heat, and let cool for 5 minutes; add tarragon.
Working in batches, transfer soup to the container of a blender; blend until smooth. Return soup to Dutch oven, and heat over low heat; whisk in sour cream. Garnish with reserved ¼ cup asparagus tips, tarragon, and oil, if desired.
Recipe by Southern Lady Magazine at https://southernladymagazine.com/asparagus-fennel-soup-recipe/