The teriaki place around the corner from my office also has burgers and chicken sandwiches on the menu. You can get a double burger or double chicken if you want, too. But there's no mix-and-match marriage of the two ideas.
That seriously bugs, but I hit a language barrier each time I customize my order, so it took about six months for me to get the nerve up to ask...
self: Can I get a double chicken burger, half beef?
I can't transcribe the span of awkwardness which followed, as it was largely hand gestures, but I was eventually able to convey what I wanted.