And while they're tilting and rocking, they're probably also skittering around on the plate, because they're not making much contact on their round bottoms.
One trick is to shave a small slice off the bottom of each egg half. That works well enough, but sometimes the yolk is too close to the wall of the egg, and there's not enough white to slice off.
Another way to thwart the rolling is to place the eggs in a container that’s slightly smaller than the eggs as they lay sideways.
Turns out that "small square" Ziploc containers I have are exactly the right size for large eggs. I fit seven eggs into a container and when the top was put on, it flattened the top and bottom of the eggs. (I'm sure there are other brands that would work, but these are the ones I happen to have.)
Later, I just had to make sure I sliced the eggs horizontally between the flattened sides, and the egg halves politely stayed in place.
There's no guarantee that Ziploc won't change the dimensions of those containers at some point, or that your eggs won't be slightly smaller or larger than the large eggs that I buy, but I'm sure you can find some container that would work.
If nothing else, I'm sure you could find something to put in the bottom of a container to take up a little space. A folded kitchen towel inside a plastic bag, for example, if you're using a larger container.
The whole point is to find something that will compress the eggs just a little to flatten the top and bottom without mashing the eggs so much that you get a weird-shaped eggs. And then just cut them correctly.