Leeks: Remove the roots and dark green ends of the leeks. Cut each in half lengthwise, then slice into roughly ¼ inch halfmoons. Transfer to a large bowl and cover with water. Swish around to remove dirt, then drain and rinse well.
Saute: Heat butter (or oil) in a large pot over medium/high heat. Add leeks, celery, and garlic, cooking until the leeks wilt down and are soft, 7 to 10 minutes.
Boil: Add the potatoes, thyme, and broth. Cover and bring to a simmer, cooking until potatoes are fork-tender, about 15 minutes.
Puree: Fish out the sprigs of thyme, then puree until smooth using a handheld immersion blender (or carefully working in batches puree with a counter top blender).
Serve: Stir in cream. Taste and add salt and pepper, as needed. Ladle into serving bowls and top with sliced green onions.
Notes
*For a vegan option, use canned full-fat coconut milk.To make this in the Crockpot, transfer all ingredients to slow cooker in step 3 and cook on low for 6 to 8 hours (or high for 4 to 6). When finished, continue steps 4 and 5 as instructed.