This is a great question, and my best Googling / Amazon-searching efforts aren't turning up much. Maybe time to write that book :)

After considering this one for a bit, I can also imagine why this would be a tough book to write. With browser technology iterating relatively quickly, and with new CSS features continuously developing, any resource that dives into specifics (specific browsers, specific features, specific issues, etc) would inevitably be out of date within a matter of years, if not months. This isn't to say there can't be great resources on designing across browsers, just that talking specifics can pose a unique challenge (although perhaps great incentive to release new editions of the book each year!).

That said, here are some resources that I've found helpful and somewhat timeless in their advice. Some are more general in their discussion of best practices and considerations, while others are more specifically issue- or browser- related.

http://www.smashingmagazine.com/2010/06/07/the-principles-of-cross-browser-css-coding/
Smashing Magazine (June 2010) - a great article that covers general approaches to creating CSS/HTML to work across browsers as best as possible - an oldie but goodie.

https://www.modern.ie/en-us/category/code-with-standards
Internet Explorer (November 2013) - somewhat ironically, Internet Explorer has a great article on fundamental best practices for ensuring you have to do as little browser troubleshooting as possible late in the game.

http://css-tricks.com/search-results/?q=cross+browser
CSS Tricks (search archive) - when I have a CSS issue I'll often turn to CSS Tricks. If you want a good overview of general cross-browser issues to watch out for, this search results page gives a good top-level perspective.

Like I said, there are lots of great articles out there covering all the issues and solutions surrounding something like IE8, but they'll be somewhat irrelevant in the coming years (e.g. https://www.tinfoilsecurity.com/blog/cross-browser-development-tips-css).

So no book recommendation, alas. Hopefully some of these links might be helpful. If not, I'm serious about writing that book :)

I would suggest online training over a book— because I can hear and view examples + when working through an online course, you can pause/search/code-along with the program. But that's just my preference. Look to teamtreehouse.com and lynda.com