Aha, I've got it figured out now, and am leaving this as an example for anyone else.
timesClicked elem = accumB (0::Int) ( (+1) <$ UI.click elem)
setup :: Window -> UI ()
setup rootWindow = void $ do
button <- UI.button #+ [ string "Clickity!" ]
output <- UI.p
getBody rootWindow #+
map element [ button, output ]
clicks <- timesClicked button
-- sink :: ReadWriteAttr x i o -> Behavior i -> UI x -> UI x
element output # sink text (show <$> clicks)
If anyone wants to follow along with my progress, I've put it on github.