Last night I tried making some rice pudding, and I'm trying to get it
similar to that made by Kozy Shack™.

I put 1 cup of steamed rice and 2 cups of skim milk in a heavy saucepan
and cooked, stirring briskly with a wire whisk, until the rice started
falling apart and the milk was getting thick like a thin gravy. I added
1/3 cup of sugar and 1/2 tsp vanilla. Beat 2 eggs in a medium bowl,
then slowly whisked in about a cup of the hot milk mixture. Then poured
the egg mixture into the saucepan, mixed thoroughly, and brought just to
a boil.

The texture is a little too egg-whitey, and the taste is a little thin.

Next time, I'll use low-fat milk instead of skim (skim was all I had
available), but what else was wrong? Do I need to use just one egg?
Did I cook it too long after adding the eggs? I think it needs more
vanilla.

I know short grain rice would work better, but I'm working with jasmine
rice or long grain rice because that's the kind that's a staple in our
house, and I want to be able to make this with leftover rice.

Thanks, and best regards,
Bob