Place 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.
Cook celery and onion in the reserved bacon drippings until onion is soft and translucent, about 5 minutes. Stir in garlic, and continue cooking for 1 to 2 minutes.
Add cubed potatoes, and toss to coat. Sauté for 3 to 4 minutes. Return bacon to the pan, and add enough chicken stock to just cover the potatoes. Cover, and simmer until potatoes are tender, 15 to 20 minutes.
Meanwhile, melt butter in a separate skillet over medium heat. Whisk in flour. Cook, stirring constantly, 1 to 2 minutes. Whisk in heavy cream, tarragon, and cilantro.
Bring cream mixture to a boil, and cook, stirring constantly, until thickened, about 5 minutes. Add cream mixture to the potato mixture; stir to combine.
Transfer about 1/2 of the soup to a blender and purée. Return to the Dutch oven. Adjust seasonings to taste.