It is possible to double move over a sanctuary. Here's how a typical game start works.
Say I go first.
1. Start the game at site 1. Fellowship phase.
2. At the end of my fellowship, I must move. [to site 2. First move to the turn]
3. Shadow/M/A/A/S/R phases. [at site 2]
4. At the end of the regroup phase, you choose to reconcile [stay at site 2] or move [to site 3].
Say that you move [second move of the turn]
5. Shadow/M/A/A/S/R phases. [at site 3]
6. At the end of the regroup phase, you must reconcile (unless the move limit is other than 2, via cards like
Rivendell Waterfall), and it is your opponent's turn.
In this case, my next turn will begin with sanctuary healing, which occurs at the start of the fellowship phase when the fellowship is at a sanctuary. So, note that if you start your turn at site 2 (bc of only moving once your first turn) and double-move, you will skip sanctuary healing.
That's probably more detail than you need, but I thought it would be best to be thorough! GLHF w/ your games!
PS: the reason to hit the concede button is that without it your opponent is stuck (for 15 minutes I think) waiting for the system to give them an auto-win.