Make a hot water bath for the flan by setting a 9 by 1 1/4-inch-deep pie pan in a slightly larger pan.
While holding down metal pie pan so it won't float, fill outer pan with just enough hot water to come up around pie pan.
Then remove pie pan and put only pan of water in a preheating 350° oven while you mix custard.
In a small frying pan over medium heat, melt the 1/3 cup sugar.
To melt evenly, shake and tilt frying pan rather than stir sugar.
Once melted, sugar will caramelize quickly; as soon as it does, pour immediately into the pie pan.
Using hot pads to protect hands, tilt pie pan quickly to let syrup flow over bottom and slightly up sides.
If syrup hardens before you finish, set pan on medium heat until syrup softens, then continue.
Beat together to blend eggs and the 6 tablespoons sugar; add milk and vanilla.
Set caramel-lined pan in hot water in oven; pour in egg mixture.
Bake in a 350° oven for about 25 minutes; test doneness by gently pushing custard in center with back of a spoon€” when done a crevice about 3/8 inch deep forms.
Remove from hot water and chill at once.
As the flan cools, caramel dissolves somewhat.
When cold, loosen custard edge with a knife, then cover with a rimmed serving plate.
Holding plate in place, quickly invert:.
The flan will slowly slip free and the caramel sauce flow out.