Step 1Trim the leaves from the celery stalks and trim away the rough bottom edges. Rinse the celery of any dirt. Chop the celery into small pieces and place in a pot.
Step 2Trim the root and top of the onion. Cut the onion in half from the stem end to the root bottom. Turn each half flat-side down. Slice from the root end to the top in 1/4-inch slices. Cut each half in 1/4-inch slices crosswise. Place in the pot.
Step 3Add the broth and bring the soup to a boil. Reduce the heat to a simmer, partially cover the soup and simmer until the celery is tender, 30 minutes. Strain the vegetables from the liquid, keeping the liquid in the pot. Puree the vegetables in a food processor or blender. Return the puree to the pot and add the half-and-half and milk. Stir to mix well, then add salt to taste. Reheat over low and serve hot. Do not let the soup come to a boil.