Bring first 4 ingredients to a boil in a large nonreactive saucepan over high heat. Reduce heat, cover, and simmer 40 to 45 minutes or until tender.
Puree until smooth in 2 batches in a food processor, adding 1/4 cup half-and-half and half the marinated artichoke hearts to each batch. Transfer each batch to a large nonreactive pitcher.
Stir in remaining 1 cup half-and-half, milk, salt, and pepper. For extra tartness, if desired, stir in 1 to 2 tablespoons reserved artichoke liquid. Cover and chill at least 2 hours before serving.
Whisk well, ladle into chilled soup bowls, and sprinkle with chopped chives.
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