I haven't seen much coverage on this anywhere aside from a tumblr post i can't find right now.
So i'm gonna try and cover this as best as i can so maybe other folk can try it out and experiment with what they can do with it!

First of all, to peform the move itself you're gonna need AT THE BARE MINIMUM have the following powerups:

1. The Dream breaker itself (duh)
   
2. Sun greaves

Keep in mind those are the only two things you are OBLIGATED to have for the technique to actually work but things like the Sunsetter help make it easier to do.

To peform it is quite simple:

1. Throw your weapon into the air like you'd normally do;
   
2. Jump into it while it's still airborne;
   
3. Kick it mid-air with the Sun Greaves;
   
4. Profit!

Missile breaking has two main uses: combat and mobility

For it's mobility uses, when peformed oustide of a combat situation the Dream Breaker will shoot itself diagonally towards wherever Sybil is facing, similar to how Soul Cutter works, until it hits the ground and comes to a complete stop.
If done right, Sybil will backflip away from it and gain a little bit more height and airtime off of it.

https://www.youtube.com/watch?v=pK2y2rSOUZ8


As for it's combat uses, If an enemy is within a certain radius of Sybil, the Dream Breaker will turn itself into a homing projectile, targetting enemies within a certain radius of the player's position and dealing high amounts of damage to the receptor.

(Keep in mind that your magic gauge does not fill whatsoever when using this attack against any enemies.)

https://youtu.be/E1u8i5g1CIk

Positioning and timing are key!!!

Keep in mind that if Sybil's body makes contact with the Dream Breaker she will pick it up, even when airborne, to successfully peform it you want to hit it with the greave's hitbox and only the greave's.

Some people might find it somewhat hard to do it consistently, what i usually do to help when i'm training is:

1. Throw it while standing still on the ground;
   
2. Jump into it when it's almost at the apex of it's trajectory;
   
3. As soon as Sybil reaches the apex of her jump, kick with the greaves.

(throwing your weapon after a Sunsetter jump and keeping your camera facing Sybil's side profile can help out with timing the jump and kick too!)
It might not work for everyone, so keep experimenting until you find a strategy that best suits you!

And... that's it! All you need to know about the basics of Missile Breaking!
Have fun using it!