Ice Water Salad

Thinly slice vegetables on a mandoline (they should be about the thickness of a nickel). Fill a large salad spinner (or bowl) with ice water and add vegetables; let sit at least 15 minutes or up to 6 hours. (Cover and chill if letting sit longer than 1 hour.)
Meanwhile, preheat oven to 325°F. Toast walnuts on a rimmed baking sheet, tossing once, until deep golden brown, about 9 minutes. Let cool, then crush or chop to about the size of a lentil.
Smash anchovies, garlic, and a big pinch of salt in a mortar and pestle to form a smooth paste. (Alternatively, you can use the side of a chef’s knife to crush the garlic, then mash with the anchovies and salt on a cutting board.) Transfer to a small saucepan and add 1/3 cup oil. Cook over medium-low, stirring often, until garlic is just starting to turn golden brown, about 3 minutes. Transfer to a large bowl and let cool slightly. Mix in walnuts and lemon juice; taste and season with salt.
Skim off any ice cubes from water bath and drain vegetables. Cover salad spinner and spin vegetables, periodically pouring off any collected water as needed, until they are very dry. Add to bowl with dressing and toss to coat. Add half of Parmesan and toss again just to evenly distribute.
Arrange salad on a platter. Top with remaining Parmesan, drizzle with some oil, and season with pepper.