Add olive oil to a large soup pot. Over medium heat, saute 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 saute for about 5 minutes, until the leeks have softened. Add 4 cups of water and vegetable bouillon. Boil on high for 10 minutes, then add the broccoli. Continue to simmer at a low boil for another 10-15 minutes.
While your soup is cooking, make your leek garnish. Slice the leeks lengthwise, creating thin 3-4 inch strips. Rinse the leeks well, and while still wet, add to a small dish, and cover with buckwheat flour. Try to mix the flour in well, but don't worry about evenly coating. Add enough olive oil to the pan to partially submerge the leeks. Once the oil is hot enough for frying (it should pop when you add a strip of leek to the pan, but not burn it), add the leeks to the frying pan. Cook until golden brown, and then let drain on a plate covered with paper towels or a brown paper bag.
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 idea is to have a nice pureed soup, but not make it super smooth.
Add the pureed soup back into the pot, and add another cup or two of water until it reaches the right consistency. Season with cracked black pepper. I didn't need to add extra salt, but that is also a seasoning option.
Garnish wish fried leeks, or enjoy as is!