Hanging a large bowl on a stone backsplash
I would find a hanger you like along the lines of this one that's deep enough for your bowl: 



Then I'd hang it from a screw placed inside or behind your range hood. I'm assuming that it's a sheet metal hood with a hollow underside. 

If the bowl is too high, use a decorative chain or rope to connect the top of the hanger with the screw at the desired height.