Both are grammatically correct, with no difference in register. You can use either. The second construction sounds a bit more natural/logical perhaps (see Kate Bunting's comment).
For reference, see Ngrams: this, this, this - but there aren't a lot of hits, and there's a word limit to the string(s), so these results aren't completely reliable.
The entry for experience in the Cambridge Dictionary contains two definitions:
[ U ]
(the process of getting) knowledge or skill from doing, seeing, or feeling things
[ C ]
something that happens to you that affects how you feel:
The first one is the kind of experience that relates to work. There is a [U] next to this entry: this indicates that is ...
It is the usual copular verb, in a rather strange form.
This is a subjunctive passive construction - rather rare. The fact that "put" is an irregular verb also hides the passive construction.
The executioner puts the prisoner to death. / Active
The prisoner is put to death (by the executioner) / This is passive is+ past participle. The past ...
Be is present subjunctive:
The subjunctive is most noticeable with the common but grammatically complicated verb be. In the present subjunctive, be staunchly remains be instead of changing to am, are, or is according to its subject. (M-W)
It is triggered by the verb ordered:
The other uncontroversial use is in sentences like the formal and often ...