As, Riley's answer plainly states, there is currently no way to view your stats from brawls. A quick google search will show that there are plenty of people asking for this feature and so far, Blizzard has been pretty good about listening to the players, so this may be a feature that we see some day.
To offer a potential explanation as to why this feature was not included, there have been 14 brawls announced so far and potentially more to come. In order to show stats for brawls I see a couple options for presenting the data, none of which are very good:
- Individual stat sheets for each brawl
This could get bulky quick if they add many more brawls over time and for the game types like We're All Soldiers Now, the page would be very bare and uninteresting given how they currently show stats, for example, you would only have one bar and then a big drop-down like the professionally edited image below
Original image without my additions.
Either that or you need a different screen layout just for brawls, which is not good from a UX perspective.
- One nonsensical page that wouldn't tell you very much.
For example, you would see stats for McCree from High Noon (headshots only) and The Good, The Bad, and The Ugly (not too different from public quick play) and Arcade (more health, short cooldowns, faster ult charge and short respawns). This would become more convoluted if they add more brawls, like types where they drastically increase/decrease damage or healing.