Michael

The Black Burst Dragon was Rose's vassal dragon. She was with it for over 10,000 years, and has a soft spot in her heart for the poor thing.

Baby dragons: Obviously dragons reproduce! On the moon it's shown that Rose cared for baby dragons, possibly Micheal's own offspring. Sadly they didn't survive to the present age.

When Rose met Micheal for the firs time, she was with Zeig.

first meeting

Rose first met Michael after she had recently obtained her Dark Dragoon Spirit. In order to gain a Dragoon Spirit, the dragon must die. Presumably, the Black Burst Dragon either had offspring already, or reincarnated itself after death in the form of a baby dragon. Rather like a Phoenix is said to arise from it's ashes, the new Black Burst Dragon arose from a cloud of darkness after Rose spoke to it.

Script:
Rose: Darkness dragon...sorry, we don't have time. Winglies are mighty, mighty enough to annihilate us, and the entire species. In order to liberate Humans, we have to sacrifice you, Dragon. Now is the time to decide, extinction or war.

Zeig: We choose war.

The Dragoon Spirit left by the Darkness Dragon, and this newborn dragon will be the spearhead of the war.

(Micheal is a cute little baby at the moment.)

Rose: This baby needs a name.

Black Burst Dragon, Michael. How about Michael?

Zeig: Rose's Vassal Dragon, the Darkness Dragoon, the Black Burst Dragon, Michael. It's a good name. So now, we can go.

Rose: Yes.
The real war has not started yet.

death

We're not sure if Michael really died, or if he was already dead. The Moon mixes reality, regret, and the past. So Rose may have already killed Michael in a similar situation and never come to terms with it. Michael also may have simply run off, or  Rose may have left him somewhere. We really don't know.

But when on the Moon, the Black Burst Dragon attacks the party, and Rose must face it. Rose is horrified that Michael doesn't recognize her. However, she and Dart must defeat the dragon, because as she explains, it's gone rouge.

After Battle Script:
Rose: A dragon's body is capable of enormous endurance. But, it's heart is not. After many fights, it becomes savage, and starts to blindly attack people and in the end, it won't listen to orders of dragoons either.

That's why I killed Michael!!

I killed him with these hands!

Dart: That's enough looking back on the past.

You lived for 10,000 years because you didn't want to see the future be destroyed, right Rose?

Rose: Thank you, you reminded me of what is important.