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1
Fill large bowl with ice water and juice from 1/2 lemon, then drop same lemon half into water. Cut off artichoke stems and discard. Using serrated knife, cut top inch off artichokes. Working with 1 artichoke at a time, bend back outer leaves until they snap off close to base. Discard several more layers of leaves in same manner until exposed leaves are pale green at top and pale yellow at base. Cut off green tips. Gently pull open center of artichoke and, using melon baller or spoon, scoop out small, purple-tipped leaves and fuzzy choke. Using sharp paring knife, trim outside of base until no dark green areas remain, then quarter artichoke, rub cut surfaces with other lemon half, and drop into water. Trim remaining artichokes in same manner.
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2
In heavy 6-quart pot over moderate heat, heat oil until hot but not smoking. Add onions, leek, and garlic, and saute, stirring occasionally, until soft, about 8 minutes. Drain artichokes and add to pot along with potato chunks, chicken stock, salt, pepper, and 4 cups water. Bring to simmer, then reduce heat to low, cover, and cook until artichokes and potato are soft, about 20 minutes.
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3
Working in 4 batches, puree soup in blender until very smooth, then pour through fine-mesh strainer into clean pot. Set over moderately low heat and stir in juice of remaining lemon. Ladle into shallow bowls and gently place nettle-walnut pesto crostini in center of each.
