That does not really answer my question. But it does not matter. US immigration authorities are not tasked with, nor do they have the authority or jurisdiction to determine, whether you are eligible for an ITAR license. Immigration officers only determine whether the job is a specialty occupation and whether you have the appropriate licenses and education/experience to fulfill the position.
So whether an employer can request a license for you to work on such projects is not an immigration question. They need to contact the appropriate licensing authority.
As far as the new employer filing for you, if the job requires a license, then you must get the license in order to qualify for the new employment on H1B. Then they file the H1B petition on your behalf. This is done exactly the same way as any original filing, except that if you were counted toward the cap at least once within the previous 6 years, you will not be cap-subject again.
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