You really need to look at the snow level forecasts and make a best guess, as there can be pretty wide fluctuations year by year. It also depends to some degree on how comfortable you are spending a lot of time in snow, plus the creek crossings can be even more dangerous for the earliest into the Sierras. This past year was really, really mild/easy. The year before was, I believe, more difficult than average. I walked through there on about an "average" year I think (2008), if anything a tad easier than average. I started into the Sierras in early June and things were fine.
There are just a whole lot of variables, certainly to include your average pace, it's hard to say much here. If you're trying to push towards "earliest" date, however, I think it's important that you think ahead of "flipping" strategies, i.e., hitch or bus or in some cases even walk around areas that can't be done, then if you didn't do a walk-around, flip back later to pick up the missing piece(s). If, of course, schedule and various life-issues allow.
The other important thing is to have the experience and wisdom to know when to turn back or turn off on to a side exit route and bail.
Sorry to be so vague. Have a look at the "snow conditions" part of postholer.com.
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