Add the quartered potatoes to a large pot and cover them completely with water by 1 or 2 inches. Place a lid on the pot at an angle and simmer over medium heat for 20–25 minutes until tender, or until you can pierce them with a fork.
Drain the potatoes and set them aside to cool.
Make the dressing by draining the cashews and dates (discard the soaking water) and placing them into the blender, along with the new water, nutritional yeast, apple cider vinegar, yellow mustard, garlic, and salt. Blend until totally creamy and smooth, about 1–2 minutes. Set aside for now.
When the potatoes are cool enough to handle, cut them into bite-sized pieces and place them into a large mixing bowl along with the celery, onion, parsley, dill, and all of the dressing from the blender.
Gently stir until everything is combined. Taste for seasoning. Chill for at least an hour before serving.