Add ice water and your sliced veggies into the salad spinner. Keep them in the fridge for an hour. If you are short on time, keep them in the ice water bath for at least 15 minutes. You can keep them in the ice water for as long as 6 hours.
Meanwhile, use a mortar and pestle to make a smooth paste of the oil-packed anchovies and garlic. Put this paste in a small saucepan along with some salt and oil. Cook this mixture until the garlic turns brown for about 3-4 minutes. After that, transfer this dressing to a large bowl.
Take a rimmed baking sheet and lay the walnuts on it. Roast the walnuts for about 9-10 in an oven that has been preheated to 325℉. When they are roasted, let them cool and then chop them into small bits.
Add the chopped walnuts to the bowl with the dressing. Also, pour lemon juice over it and mix.
Now, take the salad spinner out of the fridge and drain the water. Then, spin the vegetables and dry them out completely.
Add the vegetable slices to the bowl and toss them so that they get covered with the dressing perfectly. Add half of the parmesan and toss again.
Now, serve the salad and top it with the remaining parmesan, a drizzle of olive oil, and a sprinkle of black pepper.