Your initial sentence is definitely acceptable and easily understood, but technically not quite right.
That's because this structure (the more..., the more...) allows for showing that two things are directly proportionate to each other i.e. an incremental increase in thinking causes a corresponding incremental increase in how puzzled you are. So, if you think about it for two minutes, you may be puzzled x2, and if you think about it for four minutes, you may be puzzled x4.
However, when you remove one instance of 'the more' from the sentence, the direct implication is that the change (increase) happened only once.
In grammatical terms, this structure uses two parallel clauses containing comparative adjectives to establish proportionate change. Removing one 'the more' from the sentence would alter this structure because one adjective is no longer comparative and, hence, the clauses are no longer parallel.
An example would be the popular phrase, "The more, the merrier!" Applying the logic you've used by removing one 'the more' to this sentence would transform it to, "The more, the merry." The grammatical error is a lot more obvious here.