punctuation - Simultaneous comma and dash use

There are a lot of posts asking if it's okay to mix commas and dashes like this:

I walked my dog – who is nice, around the neighborhood.

And that's definitely not correct.

But what about the case of using the dash and comma simultaneously?

Which of these is correct?

I walked my dog, Frank – who is nice, around the neighborhood.

I walked my dog, Frank – who is nice – around the neighborhood.

EDIT:

The sentence contains incidental information within incidental information.

An appositive within an appositive. This is equivalent to:

I walked my dog [Frank (who is nice)] around the neighborhood.

However, when using dashes and commas, I've read I can't do this:

I walked my dog, Frank – who is nice –, around the neighborhood.

So, I have to close two, "brackets," with one punctuation mark: a dash or a comma.

The lack of a "closing" dash troubles me just as much as the lack of a "closing" comma.