The browse control has two components, the actual browse button:
{F374069}
...and the browse dialog:
{F374071}
You can get to examples of both via {nav Differential > Edit Revision}.
These are both kind of ugly.
The browse button should probably be integrated into the tokenizer, like the mock in T5750, and just be an icon instead of an icon + text. I wasn't immediately able to get this working cleanly with bare PHUIButtonView.
The dialog probably just needs some tuning and cleanup.
(The dialog title also needs to use an actual human-facing string.)