My family including a 16 year old and a 20 year old will be taking a 3 day hike from Pinkham Notch to Lake of the Clouds next week - June 24, continuing on to Mizpah ending at Crawford Notch. This is our first family backpack trip in the White Mountains. In past years we have hiked 6 -12 mile day hikes in Glacier, the Cascades, Smokies as well as shorter hikes in the Blue Hills and Acadia.
I would love feedback on which trail to take our first day; Boulder Glen or Boott Spur Trail - terrain, challenges, etc. At this point we are inclined not to take Tuckerman's Ravine Trail. How likely is it to encounter snow on these trails at this time?

Also, we are considering after getting to Lake of the Clouds, to unload some supplies and head out to the summit of Washington before dinner.