This year early June is good for the Sierra. The JMT northern section allows 4-5 day resupply BUT, with three people, I think you are a bit too late to get permits. If however you were to start from Twin Lakes (out of Bridgeport), you could likely get a permit and could intersect the JMT at Tuolumne Meadows. As for fires, there are no fires allowed on a lot of the JMT due to elevation and this year as a drought year, there is likely to be a ban on all open fires.

Early June is too early for the Rockies. They are getting more than normal snowpack this year. Maybe some sections of the CDT in New Mexico? Lots in Canyonlands, Zion, etc but not sure you could put together a continuous 2-week trip. Grand Canyon - same permit problems. Anywhere poplular that requires permits you are a little late in planning.

Someone else will have to chime in on early June in the PNW.