To prep the leeks, slice off the root end and halve leeks lengthwise. Slice away the tough dark green leaves and discard or rinse and reserve for stock. Immerse in halved leeks water, peeling back the leaves and swishing or rinsing out any sand. As you remove leeks from the water, rinse well under running faucet. Pat dry, slice away any remaining dark green parts, then cut white and light green parts into thin slices.
Fold in the rutabaga and celery root. Add broth and thyme leaves, and bring to a boil. Turn heat to medium-low and cover with lid slightly askew. Cook at a bubbly simmer until roots are very tender, 15 - 20 minutes.
Remove the thyme sprigs, and, working in batches, puree the soup, then return to the pot. If needed, add up to 1 cup of water to thin the soup as it blends. Fold in the cream. Add salt and pepper to taste. Rewarm over low heat, stirring constantly.