Eric,
I can't really speak to the La Pavoni--but that won't stop me

--but I had to come up with a totally different technique to pull good 7 g singles on my La Peppina that's completely different from 14 g doubles.
With singles I threw out any attempt at a long pre-infusion and multiple partial pulls and ended up with a quick partial pump and go otherwise the shot was over-extracted. If your getting good singles, but thin doubles, maybe you should try adding Felini pulls (partial pulls) and extend the pre-infusion time before pulling the shot to get a fuller extraction.
You can see the different techniques I use on the
Spring Lever Video thread. Hopefully some of this translates to a La Pavoni.
I went from struggling to pull a good single to now preferring a single to a double, though my doubles are still very drinkable.
This
La Pavoni Instructions thread talks about Felini pumps and pre-infusions, you might pick up some more relevant tips there. Also, I think there's a real art to doing a second pull without fracturing the puck and getting channeling leading to thin, watery shots... with the La Peppina's flapper valve, this isn't much of an issue.