I was wondering if there is a known graph product which takes graphs $G$ and $H$ and produces an output graph with the girth of $G$ but the degree of $H$? 

Kind of like zigzag product which produces a graph the size of $G$ but degree of $H$; however, the zigzag product will have the girth of (at most the girth of) $H$ (I believe) so this is no good for my use.

Thanks in advance