they are the shank on a circle/shank hanger for half round gutters
Must be the picture angle but it looks like you only have about an inch/ inch and a half headlap for the starter course and maybe two for the field courses.
We've gone to cutting mod bit in strips for the starter shim. Two layers. Won't dry out.
A layer of felt over the I&W. Keeps the slate from embedding at top edge in the ice and water.
The new "euro" imported gutters are much better (and easier to install) than the old Bergers. I have not tried the ClassicGutters. But it looks like you have to match the original gutters on that one. I'm guessing you have quite a few repairs to do on that crappy spanish slate.