If I were white, I would break the mirror at move 12 (if not earlier, see below), as I think move 11 is black's first 'obviously' bad move. The problem is, it isn't very important...it's not the widest space, it doesn't threaten much in the corner, it doesn't take much territory, it's generally not very interesting.
Since both players have been playing a moyo style game, with high stones and big frameworks, the best move is probably a moyo move. I don't know what's best, but the obvious move is simply tengen - the boundary between two identical moyos. More reserved might be a (high) shimari from the other 4-4, or some other moyoish thing, but when both moyos are so massive and span the whole board then it seems unlikely to be interesting to step back from the boundary.
Edit: Actually, I think all the moves after move 10 are bad

. These long knights moves feel like they take territory, but actually leave loads of invasion points...at the end of your example, for instance, both players probably haven't even defeneded their 3-3 invasions! This is the classic mistake of playing in the middle of an extension from the corner and ending up neither strengthening the corner nor necessarily connecting to the extension in a useful way. In a moyo game like this, why are you even worried anyway? It's more interesting to play in the centre and look forward to attacking white's invasion.
Edit 2: On reflection, I think moves 9 and 10 are bad too, though less so. Both players have played a framework game, but then play a boring low corner shimari and give their opponent time to finish their moyos. Tengen would still be the obvious move (and is by far the most popular in pro games).
It seems the immediate key point for you to take in is
follow your plan. If you've played a big moyo framework, don't forget about it and make a little territory instead...especially if your opponent has an equally big moyo! In that case, some point between the moyos will strongly aid either of them and is generally by far the biggest point on the board.
Up to move 9, this opening has been played quite a few times by professionals, with telling continuations:
- Click Here To Show Diagram Code
[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . O . . c . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . a . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . b . . . X . . . . . X . . . |
$$ | . . . . . d . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]
The most popular continuations are a (28), b (10), c (5) and d (2). Perhaps we should make allowances for Takemiya single-handedly biasing this, having played 20 of those first 28, but either way it's clear that making a little territorial shimari is less important than playing something that works well with the rest of the moyo.