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1
Cut top half off 1 artichoke; rub cut ends with one lemon quarter. Break off outer leaves until only pale green leaves remain. Trim and peel stem. Rub with lemon. Repeat with remaining artichokes. Place lemon quarters and next 7 ingredients in heavy large pot. Add trimmed artichokes. Add enough water to cover artichokes by 1 inch. Bring artichokes to boil uncovered; remove from heat. Let stand just until tender when tester is inserted into center of artichokes, about 25 minutes. Using slotted spoon, transfer artichokes to plate to cool. Halve lengthwise. Scoop out choke and remove any spiny leaves from centers. Cut each half into 3 wedges. (Can be made 1 day ahead. Cover and refrigerate.)
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2
Bring garlic and 1 cup olive oil to simmer in heavy small saucepan over medium heat. Reduce heat to medium-low and simmer until garlic is very soft, stirring occasionally, about 30 minutes. Strain garlic, reserving oil and garlic separately. (Can be made 1 day ahead. Cover and refrigerate chopped garlic and garlic oil separately.)
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3
Add celery root to large pot of boiling salted water; cook until crisp-tender, about 1 minute. Using slotted spoon, transfer celery root to strainer. Rinse under cold water; drain well. Return water in large pot to boil. Add carrots to pot; cook 30 seconds. Add onion; cook 30 seconds. Add celery; cook 30 seconds. Drain vegetables. Rinse under cold water until cool; drain well. (Can be made 1 day ahead. Cover and refrigerate.)
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4
Puree 1/4 cup extra-virgin olive oil and walnuts in blender until smooth. Add both mustards and vinegar; puree until smooth. Gradually add remaining 1/2 cup extra-virgin olive oil and 1/2 cup reserved garlic oil. (Can be made 1 day ahead. Cover and refrigerate.)
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5
Mix artichokes, roasted chopped garlic, celery root, carrots, onion, celery, 1 cup dressing, chives, and parsley in large bowl; toss to coat. Transfer to platter and serve. (Can be made 1 day ahead. Cover and chill. Let stand at room temperature 2 hours before serving.)
