PCB, Plated hole with annular ring on only one side
Well as comments have said only your fab house can tell you for sure.  You should try to keep a close relationship with them as they're basically your partner in all your designs :)

Now this question just happened to come up for me as well a few months ago on a really dense board.  I use Via Systems when in the US, a very large and capable manufacturer.  I asked them a similar question, could I make vias that only had an annular ring on one layer.  Their reply was no don't do that, we can't plate the via properly that way.

Your shop may tell you it's ok but I would expect they won't like it.  There's always surface mount connectors.   Or and you'd have to ask about this to but I imagine you could just not plate those vias and only route to the top layer where the pad is.  There really wouldn't be much holding the connectors on the board at that point and I'd look at mechanically securing them to the board.  Maybe epoxy or screws or something like that.