Halstatt and Eagles Nest in One Day

Not if you really want to see them.

If all you want to do is add them to a list of places you can say you've been to, maybe.

Are you familiar with Google maps?
You can use that to search the estimated drive times from Salzburg to Eagles Nest to Hallstatt and then back to Salzburg.
I would factor in how much time you think you need at each destination to determine if it is possible to visit in one day

This is not feasible, if you want to spend more than 10 minutes at either place.

Very simple answer: no, you cannot do both in one day in any meaningful way.

But you could do Eagle's Nest and Königssee, an equally beautiful place IMHO, in the same day.

Thanks, then we will keep to our plan of a day for each. Just trying to save some time.