"become puzzled" is a correct and usable phrase in written English.
It can be used to describe a state of confusion or uncertainty. Example: As the instructions became more complicated, I could see my students starting to become puzzled.
A man who had this pretty regularly became puzzled because some mornings his check was 35 cents and some times 40 cents.
He became puzzled as to why more biologists weren't studying what he views as "the major unsolved problem" in their arena.
In the town the doctors had become more and more puzzled by new kinds of sickness appearing among their patients.
And Ms. Ip became even more puzzled when he walked her into the deli where they had first talked, five years before.
Because the authors were clear about their experimental design and the lack of data analysis in their paper, we became even more puzzled to understand the reasons or arguments that led to the eventual acceptance of this manuscript.
That first experience became one of many, and while I've become more accustomed, I remain puzzled by discrimination coming from someone who belongs to a group so frequently discriminated against.
The dénouement of frozen fates, in the novel so devastating, so majestic in its pain, in the movie becomes merely puzzling; Liam Neeson's broad face in the final closeup looks nothing but puzzled, perhaps by the mysteries of the New England accent that kept eluding his Irish tongue.
It will become clear that Avicenna's syllogistic puzzled those who came after him, and still puzzles those today who try to work out what Avicenna was doing.
In reading some of the corporate excuses/reviews used by the corporate writers to explain why they do not like the film, I become increasingly more and more incensed and puzzled.
"I saw this dark band that really did go across a large fraction of the sky and I was quite puzzled as to why it had become visible then.
