Here is my “Hotdog” themed flair:
Notice that the Stack Overflow logo is almost invisible. Is this a bug or by design?
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I've implemented Ilmari Karonen's suggestion because it's the easiest to implement without extensive code changes: you just draw a rect of the same placement as the icon before drawing the icon. This only applies for Stack Overflow, and only for the hotdog theme.
It will be deployed in build 2870 for Stack Overflow, and build 3032 for SE.com (for network flair).
(Our internal dev instance is based on an old snapshot of Stack Overflow, which is why my rep is about 10k less than it actually is.)
Suggestion: use a bright yellow background for the site logo, like this:
The yellow is close enough in luminance to white that most site logos should be recognizable on it (the new SO logo is probably close to the worst case — it's not exactly high-contrast even on white), yet it still looks appropriately garish for the hotdog theme. ;-)