When is a problem specified on a TM contained in non-uniform classes such as P/poly? In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, they mention that the input to the TM they encode in the tiling problem must be an integer $N$ in binary rather than unary, otherwise each instance will have a different input size and hence the problem would "trivially be in P/poly" rather than NP. Hence instead they must prove the problem is NEXP-complete rather than NP-complete.  

While I understand why the proof with a binary input gives NEXP-completeness, I don't understand why if $N$ is input in unary then the problem is trivially hard for P/poly? Unfortunately the model of computation they specify is in terms of TMs rather than circuits so it's not clear to me how this relates to non-uniform circuit families.
If $N$ were written in unary, then the language would be a unary language, i.e., a subset of $\{0,1\}^*$.   Every unary language is in P/poly.  That's because a P/poly machine is allowed a separate advice string, one per length of the input.  The advice string for length $n$ could just describe whether the input $1^n$ should be accepted or not.  Or, to put it another way, P/poly allows you to specify a separate circuit per input length; so you could use the "always output no" circuit for lengths $n$ such that $1^n$ is not part of the language, otherwise the "always output yes" circuit.  See https://en.wikipedia.org/wiki/P/poly.