Place the sliced bacon in a Dutch oven and cook over medium-high heat, turning occasionally, until evenly browned, 5 to 10 minutes. Drain bacon pieces on paper towels. Drain off all but 1/4 cup of the bacon grease.
Saute the celery and onions in the reserved bacon drippings until the onion is soft and translucent, about 10 minutes. Stir in the garlic, and continue cooking for 1 to 2 minutes until the garlic has browned and is fragrant.
Sprinkle the flour over the vegetables, stirring and sauteeing until the flour has browned and smells nutty.
Return the cooked bacon to the pot and add in the potatoes. Pour in enough chicken stock to cover the potatoes. Stir in the tarragon and the basil.
Cover and simmer on low heat until the potatoes are tender and falling apart, around 40-45 minutes.
Whisk in the heavy cream. Using an immersion blender, puree the soup for a few seconds, until thicker. If desired, puree the entire soup, or just enough to thicken the soup. You can also remove 2 cups of the soup and puree it in a blender, then return to the soup.