http://www.onestopenglish.com/community/your-english/word-grammar/your-english-word-grammar-even/156431.article gives some examples of correct use of "even" as an adverb to indicate that something is surprising or unexpected:-
"She even forgot my birthday"
"He can’t even spell his own name!"
However, using "even" with "to be" results in some odd-sounding sentences:-
Adverbs usually come after auxiliary verbs, but in this case it doesn't seem like moving the adverb makes a more pleasing sentence:-
"What is even that thing?"
"What is that thing even?"
Is there a correct way those sentences could even be formed, or should "even" not even be used with "to be"?
As an additional point, these formations seem very common - especially colloquially or in writing on the internet. Is there anywhere I would be able to investigate the frequency of usage and perhaps find some early examples?
For anyone interested, I found these:-
Suggesting that the use of "even" in this context is new/wrong, rather than specifically the pairing with "to be", e,g:-