A company with over 50,000 employees has probably had a few folks who've been in the position you are in. Start with your HR resources, and ask them if they can connect you with people who've done a degree part-time.
I did both an MS and a PhD part-time, paid for by my employer. Obviously, that's different. A part-time MS is a well-trodden path. A part-time PhD is not quite so well trodden, but it's been done. (Although my adviser told me flat out that nobody finishes ... if that was meant as a challenge, it worked).
I ended up taking an unpaid leave of absence, but as I said, a PhD is different, in that there's a bigger "crunch" at the end.
In the end, whether you do this or not, and whether you succeed or not is going to depend on three people (if your large company is like mine): You, Your Manager, and Your Manager's Manager. Your first two lines of management will have to fly cover for you, and deflect criticism from above and from below. You'll need to be in a position where the expectations on your work are a bit lower, in compensation for the degree work. You'll also need to realize that you won't get great ratings, and you will probably be passing up promotions and raises for the duration.
Whatever you do, don't do a degree and bolt for another job. If you do, you're just poisoning things for the next person. If you do the degree, stay for a while and show that the company gets something out of this, then the next person won't have as much of an uphill fight. (When I started, the program I wanted to use was discontinued because of folks who went to UCB or Stanford and then immediately left the company).