I was going to suggest the Forstner bit, too, but you want to be careful not to drill the full 1” deep with it, because it’s going to be more or less flat-bottomed, with a small pilot hole in the center. So some practice is definitely called for.
There are also stepped drill bits available that might make a kind of stepped cup shape (not really broad enough to be characterized as a “bowl” shape).
You might want to consider a router with the specific moulding cutter that could create the bowl shape you want.
What you’re describing is an unusual use for a router bit. You might me able to modify this router cove bit by removing the guide bearing. The biggest one I’ve found is only 1.5 inches diameter. I’d love to hear what you figure out.