Remove and discard green tops from leeks; cut white portions into slices, and set aside.
Sauté chopped onion and chopped garlic in hot olive oil in a large Dutch oven over medium heat just until tender. Do not brown.
Stir in leeks, carrots, and next 3 ingredients. Bring to a boil. Reduce heat and simmer, stirring occasionally, 25 minutes or until vegetables are tender.
Stir in thyme. Remove from heat, and let cool 10 minutes.
Process carrot mixture, in 2 or 3 batches, in a food processor or with a hand emulsion blender until smooth.
Return carrot mixture to Dutch oven, and add half-and-half. Heat over low heat until warm.
Absolutely wonderful! Perfect just as is. It has become a family favorite!