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