Add olive oil to a large soup pot. Over medium heat, sauté garlic.
Chop the leeks in “half moons” (down the center, then slice in thin rounds). Only use the lower part of the leeks. When the leek gets difficult to cut, discard from that part up. Reserve uncut leeks if making the crispy fried garnish.
Add the leeks and potato to the pot. Continue to sauté for about 5 minutes, until the leeks have softened.
Add 4 cups of vegetable broth. Boil on high for 10 minutes.
Add the broccoli and reduce the heat to medium low, simmering for another 10-15 minutes.
Once the broccoli is tender, remove the pot from heat. Allow to cool for about 10 minutes, then add it to a blender (or use an immersion blender), and process until smooth. Add the fresh tarragon while the soup is in the blender to keep the flavor fresh. Don’t over process the soup. The idea is to have a nice pureed soup, but not make it super smooth. *If you like chunky soups, puree half of the soup and then pulse the second half to leave texture.
Add the pureed soup back into the pot, and add another cup or two of vegetable broth until it reaches the right consistency. Season with cracked black pepper and sea salt.
Serve the soup hot ganished with crispy fried leeks.