How to bake burger buns so that they expand horizontally instead of expanding vertically?
Your buns are likely sticking to the bottom of the tray. If they stick, they will expand vertically because they can't expand horizontally. They're stuck.

Dust the trays with semolina flour, rice flour, and/or fine corn meal. You can even lightly dust them with white or rye flour, but the aforementioned flours are preferable.

If you're retarding the buns, consider placing parchment paper on the tray, lightly dusting the parchment with the aforementioned flours, and then panning the buns.

You could also use non-stick Silpats or silicone mats, but that's expensive if you have to buy a lot of them.

I don't know the composition of your dough, how you are shaping them, how much you are working the dough, etc. These could contribute the problem as well. Buns are usually enriched dough (they have butter, milk, etc.) made with low-protein AP flour. They don't need a lot of gluten development or shaping. If you develop the dough too much, it'll hold something close to the original shape (in this case, of a ball of dough instead of a disc).