Is this particular statement a phrase or a clause?

The difference between clauses and phrases has been extensively discussed (here, here, and likely elsewhere). And as Dusty has said, “The short answer [is that] clauses contain a subject and its verb, while phrases do not.”

In a recent comment (found here), I noted that the questioner’s sentence—“The day I was born, Granny died”—was formed by two clauses. However, phoog responded that "the day I was born" is not a clause, but rather (I presume) a phrase.

Is this true? If I am to judge by Dusty’s short answer—that a clause contains a subject and a verb—then “the day I was born” checks out as a clause. My understanding is that “the day I was born” is a subordinate clause—or, in the words of Wikipedia, “a clause that provides an independent clause with additional information, but which cannot stand alone as a sentence.”

Unless you think they are necessary to the meaning of your answer, I do not need definitions of phrases and clauses. Rather, I am looking for a definitive answer—phrase or clause—and proof as to how or why you came to that conclusion.

EDIT: I deleted this question because I felt as though it had been answered on the original post by phoog. He responded to my comment, noting that "[the day I was born is] an adverbial noun phrase. It has no predicate, as a clause must. The clause I was born modifies the day, but the fact that the phrase contains a clause doesn't make it a clause."

However, Araucaria commented on another post of mine, asking me to reopen this question. He states, "The linked-to answer is a bit misleading (to say the least), but your question is excellent, and is a very good example for people to investigate!" That said, I have reopened the question and am curious to hear any and all opinions and/or interpretations of phoog's explanation.

Answer

"The day I was born, Granny died”

Inverted ---> "Granny died the day I was born.”

In full ----> "Granny died on the day that I was born.”

Same structure ----> "Granny died on Tuesday.”

Answer

"the day I was born" is a noun phrase.

"I was born" is a clause.