If Void Assault starts 2 Showdowns, which one happens first?

The turn player chooses which showdown to resolve first.

Here's the breakdown: Void Assault resolves as a single spell — first moving the friendly unit, then the enemy unit. Both battlefields become contested during the spell's resolution, but no showdown begins until the spell fully completes (FAQ #10632).

After Void Assault finishes, the next Cleanup stages a Showdown at each battlefield that was made contested (Rule 323.8). Since more than one Showdown is staged and the game is returning to a Neutral Open State, the turn player chooses which battlefield's showdown to start first (Rule 323.11, FAQ #10342). The first showdown resolves completely — including all combat, conquering, and triggers — before the second one begins (FAQ #7567, #4822).