1heat peanut oil in a 7 to 8 quart pot over high heat until it registers 375° on the candy/deep Fry thermometer.
2For the dressing, whisk together olive oil, grated Parmesan, mayonnaise, lemon juice, garlic, anchovy paste, Dijon, Worcestershire and Tabasco. Season with salt and pepper; preheat broiler with rack 6 to 8 inches from element.
3For the chips, cut tortillas into six wedges each and Fry in six batches until light golden brown, about 1 1/2 minutes per batch. Transfer chips to a bowl using a slotted spoon, toss with salt to season, then transfer to a clean baking sheet while frying remaining tortillas
4toss the chicken with 1/3 cup dressing. In another bowl, toss lettuce with 3 tablespoon dressing. Top fried tortillas with 1/2 cup shaved Parmesan, some of the remaining dressing, chicken, and remaining shaved Parmesan.
5Broil nachos until cheese melts, about two minutes, then top with dressed lettuce and tomatoes; serve at once. Makes eight servings