Both of the example sentences are ungrammatical.

Instead, these two will work:

"What vegetable has this salad got the most of?"

"What does this movie have: better script or better direction?"

However, I think both would benefit from using "is" instead of "has" because the focus is on a single ingredient, not on the whole. I think that's what's causing you to need these complex expressions, which are hard even for natives to get right. Compare these two:

"What is the main vegetable in this salad?"

"In this movie, which is better: the script or the direction?"

Both of the example sentences are ungrammatical.

Instead, these two will work:

What vegetable has this salad got the most of?

What does this movie have: better script or better direction?

However, I think both would benefit from using "is" instead of "has" because the focus is on a single ingredient, not on the whole. I think that's what's causing you to need these complex expressions, which are hard even for natives to get right. Compare these two:

What is the main vegetable in this salad?

In this movie, which is better: the script or the direction?