If I counter a spell with Defile and my opponent counters my Defile with Abandoned and sends it to my hand, can I let Abandoned resolve and recast my Defile on the original target?

Yes — assuming you mean Defy (there's no card named "Defile"), you can let Abandon resolve and then recast Defy on the original spell.

The chain resolves one item at a time (LIFO), not all at once. After Abandon resolves and returns your Defy to your hand, a priority window opens before the original spell resolves (Rule 340.1; FAQ #9457). Since Defy is a [Reaction] spell, you may play it during that window to target the original spell again, which is still on the chain.

One caveat: the new Defy must still have a legal target — the original spell must still be on the chain and meet Defy's restriction (printed cost of 4 or less energy and no more than 1 power). If it still qualifies, your recast Defy resolves and counters it.