Strange Alien Plant Form

We recently noticed these green vines growing in our flower garden and across our yard.  They did not look like the flower seeds with we had planted, but we had used some compost from the composter that the previous owners had left.  We though we might have some unintended squash or volunteer cucumbers growing from seeds in the compost.  That's fine.  Great!

Then, we noticed this:

This isn't a cucumber.  Not one that I've ever seen!  Any ideas?

And the vines are taking over our little flower plot (please do not judge how flithy our siding is.  I'll clean it soon, I promise).