Does $\mathsf{P/poly}$ have subexponential-size bounded-depth circuits?
What you ask for should have bad consequences but I cannot think of any immediately. So I have only some pointers to what we know.

Check out Viola's On the power of small depth computation The best we know is Valiant's construction for boolean circuits: log depth linear sized circuits to depth 3 subexp circuits. (We know better for arithmetic circuits.) 
There are also some results of Beigel/Tarui on ACC begin contained in bounded depth circuits of superpoly size. I don't recall it being extended to all of $NC^1$ though.