“Got” is a very strange word in the English language. Technically it’s the past tense of the verb “to get”. (And just to be difficult, so is “gotten” — at least in some dialects.) But that’s not the only way it’s used.
For example, the fragment “once you’ve got the window open” is fairly normal in common usage (though I’m fairly sure it’s frowned on in more formal grammar).
If you expand the contraction, “once you have got the window open” is still technically valid but is starting to sound quite dodgy — and it can be replaced by the equally-valid-but-still-dodgy “once you have the window open”, making the word “got” itself seemingly redundant. And yet “once you’ve the window open”, which logically ought to be the same, completely doesn’t work.
Of course the most correct (and formal) phrasing would have been “once you have opened the window”, which also works in the contracted form “once you’ve opened the window”. And yet it seems like most people seem to default to “you’ve got”, at least in casual conversation.
The original phrasing is using “to get” in its “achievement” sense — ie. “once you have achieved the state of the window being open”… not that anyone would normally state it in such a long-winded fashion. This is also why “have” works without it, as this is “to be” — ie. “once you have the window open” reads sufficiently close to the more formal phrasing above. It’s not as clear why it stops working once you reintroduce the contraction, though.
English is weird sometimes. Or maybe it’s just me.
(And yes, at times I’m a self-professed grammar/spelling nazi. Especially when it comes to misplaced apostrophes and homonyms (including using “of” instead of “have”). And even more especially when the person making the error should know better — eg. if formal writing is part of their job. (But clearly I have no objection to nested parentheses.))
This random musing is brought to you by the symbol ‘.