translation - て form followed by て, 「見てて」

I have heard a couple of times now the て form of a verb followed by て (or perhaps it was って?) used as an imperative. Some examples are 見てて and 黙ってて, when asking someone to look and be quiet respectively.

I am also aware of this question, but in that circumstance, the listener is being asked to "stay waiting" as I understand. However, in the case of 見てて, this interpretation seems odd when all that's needed is a quick look. (A child was asking the father to look at a toy)

I assume the phrase is still a contraction of 見ていて, but how should I understand it, and what is the difference between this and 見て? What about other verbs?

Edit

Here is an example usage of 見てて in a similar situation.

Answer

I think you have almost grasped the "tournure" and I have few to contribute, but...

I assume the phrase is still a contraction of 見ていて

That's correct. As for the example, the girl in the film says おとう、見てて. That corresponds to "Look at me (doing this), Dad."

it would mean something like "check out this website (and continue doing so for a nontrivial period of time)"? Similarly, 黙ってて would mean "be quiet, and stay quiet"?

Yes, I guess so. Of course, you need context to be more precise, but I can hardly think of other possibilities.