FOR THE SAUCE VERTE
Combine all the ingredients, reserving ½ cup of the olive oil, in a blender or food processor and process to a thick paste. Gradually add in the remaining olive oil to make a thick sauce. Taste and adjust for salt, as desired. The sauce can be made a day ahead and covered with a light film of olive oil to maintain the bright green color.
FOR THE BEIGNETS
Trim the end of each cardoon stalk. With a sharp knife, cut off the leaves and the sharper edges of both sides of the stalks. Using a vegetable peeler, peel the backs of the stalks to remove the outer layer of skin and the strings, just as you would do with a celery stalk. Cut the cleaned stalks into 3-inch-long pieces and rinse them in cold water.
Over medium-high heat, bring a pan with the water, milk and salt to a boil. When it is boiling, add the cardoons and reduce the heat to medium. Cook until the cardoons are easily pierced with the tines of a fork, about 30 minutes. Remove them with a slotted spoon and set them aside to drain. This can be done the day before making the beignets, if you want.
Peel the parsnips and cut the thick upper part lengthwise into 4 pieces to make batons. Leave the narrow lower root whole.
Peel the Jerusalem artichokes and cut them into generous ¼-inch-thick slices.
Preheat oven to 200° F.
Line a platter with paper towels.
Pour grapeseed, canola or other vegetable oil into a casserole or deep fryer to a depth of 3 to 4 inches. Over medium-high heat, heat the oil to 360° F. as measured on a deep fryer thermometer.
Sprinkle some flour on a plate.
In a bowl, combine the flour, egg yolk and salt. Remove the ice cubes from the water and discard them. Measure 1 cup of the ice water and combine it with the egg mixture, whisking all together to make a batter.
Dredge the vegetables, a few at a time, in the plain flour, then dip into the batter and slide into the hot oil. Fry until golden, turning once or twice. Remove to a platter and keep warm in the oven. Repeat with the remaining vegetables. When all are ready, serve accompanied by the sauce.