Gladys is visiting
The Remarkables, on the South Island of New Zealand.
Here's the crossword:

And now the jigsaw pieces, with letters on horizontal pieces shown in blue and those on vertical pieces shown in red:

You'll notice that
each of the jigsaw pieces has exactly one letter hanging off the end. If we read those extra letters clockwise starting at the top left, we can see that it spells out THE REMARKABLES!
What a remarkable puzzle!
As requested in comments, a bit about how I solved the puzzle:
My in to this puzzle was 8 and 12 across. Because we have a rotationally symmetrical grid, I knew that these two answers would have to be the same length, as they were right in the middle of the across clue list. I still thought of Jared Leto and Mark Hamill before I eventually remembered JOAQUIN PHOENIX (and incidentally, I remembered Jafe having mentioned in chat that Joaquin Phoenix would be a good name to build a pangram around, so I figured I was on the right track). Again, with the grid being rotationally symmetric, I knew neither of them could go in the middle row, and it seemed unlikely that the grid would have three across answers squished into the top (and bottom) row while having no across answers whatsoever in the middle three rows. Thus, I put JOAQUIN PHOENIX into the third and fifth rows, and blocked off the entire middle row, since there are no across answers in between:

Now, at this step, I did identify what jigsaw pieces would go with those two rows. Unfortunately (well, I suspect this was by design, not by fortune, but still), all of the really uncommon letters (J, Q, X) were on those pieces. If I was smarter, I could have noticed that the jigsaw pieces only had one P, which would've gotten me a C in the top left corner and probably would have helped me get 1 down faster, but as you'll soon see, I got there anyway.
Instead, what I did was start looking at the clues in between 8 and 12. I knew that three down clues would have to start in between the O in JOAQUIN and the last cell in the middle row. I immediately identified QUE as the likely solution to 9 down which fit symmetrically right into the middle of the grid, and blocked off either end. Then, I looked at 11 down; I know nothing about horse jumping, so I knew this was something I would have to look up anyway. It didn't take me long to find OXER, which fit nicely with the X on the edge:

Now, knowing that 1 down had to be ??J?, it was pretty easy to identify CUJO (shoutout to Epic Rap Battles of History for the assist), even without the C from earlier. 1 across was likely to be CIA from the jump anyway, and to parallel that, 15 across should be WAR. Again, there's more I could have gleaned from the jigsaw (only one W in the grid, for example), but I was just focused on getting the crossword filled in.

At this point, 3 across has to start from the middle column, and 4 down has to be somewhere to the right of it. I spotted YOUKNOW as a likely 7-letter answer that used up a few uncommon letters, and AMAZONS fit nicely in the spot across from it (and also gets that pesky Z out of the way):

At this point, there was enough of the crossword that I could pretty much start filling things in. LYON was the only reasonable French city, which appropriately gave NON as the down clue there. I had already been thinking of AMMO and NOOB for 7 and 13 across, and those fit in nicely on the second and sixth rows. PDF and FEST worked in the bottom left, and then it was just a matter of remembering OVID and GHEE to fill in the remaining spots. Only then did I go back and play with the jigsaw, and luckily everything fit, or I would've been confused :)