Shred enough of the carrots to make 1 cup.
Cook remaining carrots in boiling salted water until just tender, but not soft.
Drain and cool carrots until they can be handled.
Shred enough to make 2 cups.
Combine the raw and cooked carrots in a large bowl.
Blend in onion, celery seed, basil, sugar, 3/4 teaspoon salt, pepper, and cheese; set aside.
Stir flour and 1/2 teaspoon salt into hot butter in a saucepan.
Heat until bubbly.
Remove from heat and add milk gradually, stirring constantly.
Bring to boiling; stir and cook 1 to 2 minutes.
Beat egg yolks in a bowl and gradually add sauce, stirring constantly.
Stir into carrot mixture.6.
Beat egg whites until stiff, not dry, peaks are formed.
Gently fold into mixture.
Turn into a generously greased 1 1/2 quart ring mold.
Set in a shallow pan of hot water.
Bake at 350°F until mixture is set, about 1 hour.
Remove from oven and let stand 10 minutes.
Run a knife around edge of mold to loosen.
Unmold onto a heated serving plate.
Surround ring with seasoned cooked broccoli spears and fill center with parsley or watercress.