I'd rather do a loop like this:

https://ridewithgps.com/routes/12486823

Just makes logistics so much easier.
But I'm game.

No map?

I bailed, couldn't take any more tree hopping.
I have a room at the Rodeway if anybody wants to crash.
Ken Moss

Has anybody found that putting a cue sheet together helps with navigation, or is this strictly a navigate-by-gps route?  I generally have my gps put away and only pull it out when I think I'm lost (or I know I'm lost...)

Ok, let's start with the basics, what to ride?  My options are:
Fat bike
FS
Rigid

What do people like to ride on this route?  Which bike do you wish you would have brought instead?

I'm leaning towards the fatty.

You should be fine.  Southern Utah hasn't had much snow this year.  The Clayton Springs SNOTEL site is showing 16 inches; should be dry real soon.  I also saw a report from this weekend that Thunder Mtn is dry.

http://www.wcc.nrcs.usda.gov/snotel/Utah/utah.html

Update on snow:  I've ridden the local dirt road up the mountain the last two weeks.  The snow line was at 6900' last week and 7200' yesterday, so maybe 300 feet of melt a week is a good rule of thumb.  I don't know if snow melt rate is linear?  It has been warm and dry in Salt Lake.  So by May 20, you might be good up to 8400'?

Yes, UT-35 would be fine.  UT-40 then down thru Provo Canyon might be better, then ride the train from Provo to Ogden, skipping the city.  Or bypass Salt Lake Valley and come out by Logan would be good too.  I can give route advice in northern UT if you need it.

BTW, here's my notional route to connect AZT to GDT:  http://ridewithgps.com/routes/4044441
Haven't ridden much of yet, but hope to start tackling sections this year.

Glad to help out.
A couple things to keep in mind:  snowpack varies greatly year to year, and there's a big difference between May 1 and June 1, a lot of melting happens then.  Snowpack has been really low this year until the past two weeks,  should be a below average year.  Check snowpack here:  http://wernersoftware.com/weatherpages/utah_snotel.php

Here's what I think for an average year on June 1:
-under 7000' should be dry everywhere
-7-8k' would be mostly ride-able, may hit some snowdrifts/mud that you'll have to hike thru
-8-9k' miserable slog thru snow and mud
>9k' bring snow shoes

There's no way you can bike Wasatch Plateau Skyline June 1. July 1 would be iffy, but probably OK.
Same with Soapstone & Cedar Breaks, but they're a bit lower and should be dry July 1.

I would advise starting July 15, or find alternative routes that stick to lower elevations.  All the paved roads will be open by June 1, so you could bypass the high elevation trails if they're still snowed in.

Anything above 8000 feet is going to still be buried in snow in May.

Last fall I rode Bountiful B/Skyline to Big Mountain.  There was a bit of hike-a-bike up towards Grandview, but not too bad.  The trail from Grandview to Big Mountain was generally atrocious; with a bit of trail maintenance it would be an awesome ride.  It's a beautiful area. 
I've looked at maps a lot over the last few years, looking for good dirt tours in the Wasatch; I need to get out and ride/explore more.  Hooking up to Wasatch Plateau Skyline from SLC looks very do-able.