Step 1Remove the leaves from the artichokes, then scoop out the choke leaving the bottoms. Discard the leaves. Cut the bottoms in half. Put them in cold water with 2 tablespoons of lemon juice until ready to use, then drain and pat dry.
Step 2Pour the olive oil into a 6-quart saucepan. Saute the onions until soft, about 4 minutes. Add potatoes and carrots; cook about 5 minutes.
Step 3Add the artichokes, 4 1/2 cups hot water, the dill and salt. Cover and bring to a boil over medium heat; then simmer until the potatoes, carrots and artichokes are fork-tender, about 12 to 15 minutes. Do not stir, but occasionally lift and swivel the pan back and forth to move the mixture and keep it from sticking to the bottom of the pan, being careful not to break the vegetables. Reduce the heat to a low simmer while preparing the avgolemono sauce.
Step 4To make the avgolemono sauce, beat the eggs well for about half a minute in a small bowl with a fork or wire whisk, then slowly add the remaining lemon juice. Add the dissolved cornstarch and then slowly add about a cup of broth from the cooked mixture. Pour the avgolemono sauce slowly over the vegetables and cook over a low flame for about 3 minutes. Remove from the heat and let stand for about 10 minutes before serving.