One thing about cutting thin slices or just even slices is to let the meat get really stiff but not frozen in the freezer before cutting.
As I described, we will freeze the meat when we butcher the deer, separating the bigger roasts from the rest of the "grind meat". When we are ready to process it, we will take the grind meat, let it thaw enough to cut, then cut it into pieces small enough to fit in the grinder.
The roasts, we will work the same way, thawing them enough to slice, which lets you make really thin slices. You do have to stop once in a while to thaw your fingers.
For me, it is easier to freeze the meat first, then thaw it rather than freeze it enough to slice.