Cook the bacon in a large skillet over medium heat until crisp, about 8 minutes. Drain on paper towels.
Pour 1 tablespoon of the bacon drippings into a large pot over medium heat; add the butter. When the butter is melted, add the celery, celery root and shallot. Cook, stirring, until the shallot starts softening, about 5 minutes. Add the thyme, celery seeds, 1 teaspoon salt and a few grinds of pepper; stir to coat. Add 6 cups water and the half-and-half; bring to a simmer and cook until the vegetables are tender, about 20 minutes.
Working in batches, transfer the soup to a blender and puree until smooth, adding some parsley and tarragon to each batch (or puree in the pot with an immersion blender). Return the soup to the pot and bring to a simmer; thin with water as needed. Season with the lemon juice, salt and pepper. Top each serving with the bacon.