Put both papers in the arxiv. Have the second cite the first, if it depends on the first. You may have the first cite the second as a "preprint" as a motivation, if it's useful. This can be updated by re-posting the first paper once you know the "coordinates" (arxiv number) of the second.
If the papers form a clear sequence, you may wish to submit both papers together to the same journal. Explain why you are submitting two papers and their relation to each other in the cover letter. Otherwise submit them separately.
Tl;dr: I don't think the math publication ecosystem has a perfect solution to your problem.
First, note that even moderate-length papers can take years to be published (my personal record is of 5 years from submission to print, involving only one submission, for a 15-pages paper). If your employer only considers published papers, and not accepted or arXiv posted ones, you are basically out of good options (of course if any journal where to count, you could find a quick predatory journal, but I would not advise this).
The usual process is to post the complete paper on the arXiv and submit to a suitable journal (Memoires de la SMF is another good available option, similar to Memoirs AMS, and I think Documenta is able to publish long papers). Publishing an announcement has become rare, but is still possible (several venues for this have been mentioned already). However, when you say that proofs are thoroughly check, that is usually not true; and if it were, then you announcement would also take much time to be accepted, since the review time would be long.
So, it somehow seems that your main concern is about the accepted-to-published delay. I thus advise you to have a look at the AMS data on delays (published yearly in the Notices), you might find a good venue. Also, you could have a shot at an electronic only journal where this delay can be much shorter. Forum of Mathematics (Sigma or Pi) could be an option if your paper is very strong, but I think they now charge authors, and you will have the usual problem that you could spend a year or more waiting for a report which could turn out negative.
We lack a math megajournal which would publish any good paper (good meaning good enough to deserve publication) electronically, avoiding unecesary delays in resubmission along the prestige ladder and in finding room into issues of fixed length. That would be about the best option for you, where it to exist.
To finish on another note: once your large paper is written and waiting for a venue, you may consider working on smaller scale projects to get your beans to be counted. I do not like to advise this, but given the incentives you mention that might be the only solution. It is a bit like a painter working for hire in advertisement and buying himself time for his masterpiece; be careful that your smaller scale project have some interest instead of being noise in the publication record.
Best Answer
Informing specialists about a new manuscript posted on the arXiv is a common practice. Once it is posted there is no need to send a file.
But on my opinion, asking explicitly a person you do not know to read and comment is too intrusive and impolite. You can write something like this: "I would like to bring to your attention the new preprint... I would appreciate any comments". Or "I will be grateful for any comments".