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Make the dressing
Combine pickles, anchovies, and garlic cloves in a food processor; pulse until finely chopped, about 4 seconds. Add mayonnaise, Parmesan, mustard, and Worcestershire; pulse until smooth, about 5 seconds. Transfer to a small bowl; cover and refrigerate until ready to use.
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Make the wraps
Combine chicken, lettuce, croutons, Parmesan, pickle chips, dill, and parsley on a large cutting board; chop everything together into small pieces. Drizzle with 1/4 cup of the dressing and continue chopping until well-combined. Transfer to a large bowl and toss with remaining 1/2 cup dressing.
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Working 1 tortilla at a time, lay tortilla flat on a clean work surface. Add 1 cup of the chicken mixture in center of wrap. Fold left and right edges in towards the center, then tuck the side closest to you up and roll it tightly. Secure it with a toothpick, if desired. Repeat with remaining tortillas and chicken mixture. Cut wraps in half or serve them whole.