This sentence is an example of Conjunction Reduction, the syntactic rule that deletes repeated material in conjoined constituents, for example
- Bill washed the dishes and Bill swept the floor. ➝ Bill washed the dishes and swept the floor.
- Bill washed the dishes and Bill dried the dishes. ➝ Bill washed and dried the dishes.
The relative clause modifying project in the original sentence is the focus, so let's get it out of a subordinate clause and see what it looks like:
- I admire and am very interested in the project.
which comes from
- I admire the project and I am very interested in the project.
by a perfectly normal application of Conjunction Reduction.
There's nothing grammatically wrong with this sentence.
One thing that may make it feel wrong to some -- but not others; there's a lot of individual variation here, since everybody makes up their own internal rules, for their own reasons, about what "sounds right" -- is that the first verb of the conjoined VP (admire) is uninflected for person and number, while the second verb (am) is inflected, for first person singular present tense.
Both verbs agree of course with the same subject, but morphologically instead of syntactically, which may produce some distress to those who require more grammatical parallelism between conjoined verbs.
Another related difficulty might be that the inflected form am is so closely linked to its subject pronoun I that it is difficult to separate them, and indeed most of the time they're contracted to I'm. This makes am feel rather isolated out there.
Again, this isn't a grammatical problem per se, but it can occasion some distress in some readers.
I say "readers" because nobody would say such a sentence, of course. We'd say I'm instead of am, by repeating the subject -- and adding no new syllables, so timing isn't affected. This is allowed syntactically because Conjunction Reduction is an optional rule applied to reduce unwelcome repetition, and in any given case this repetition may simply not be unwelcome.