Graph product which produces a graph with girth of one of the input graphs and minimum degree of the other

Question

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 minimum degree of $H$?

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

Thanks in advance