Noun Phrases that are Possesive -- Possible?

http://en.wikipedia.org/wiki/English_noun_phrase

For example, if I wanted to say that the neighbor next door has a cat who is sweet, do I say:

"The neighbors next door's cat is sweet." -or-
"The neighbor's next door cat is sweet."

"The parlor down the street's pizza is fantastic" -or-
"The parlor's down the street pizza is fantastic."

Or is this even possible in proper, written grammar?

Thanks!!