In a saucepan, cook and stir the water and the 1/3 cup sugar until sugar dissolves. Bring mixture to a boil; boil without stirring until mixture turns golden (about 10 minutes). Divide syrup among six warmed 6-ounce custard cups; tilt to coat bottoms. Let stand for 10 minutes.
In a bowl, beat the eggs and the 3 tablespoons sugar. Stir in the milk and vanilla. Pour into custard cups.
Place custard cups in a 13x9x2-inch baking pan. Place pan on an oven rack. Pour boiling water into baking pan around custard cups to a depth of 1 inch.
Bake in a 325 degree F oven about 35 minutes or until a knife inserted near the centers comes out clean.
Carefully remove pan from oven; remove cups from pan. Cool on wire rack; chill in the refrigerator.
To serve, loosen custard from cups; invert onto individual serving plates. Makes 6 servings.