I'm playing a dragonborn and was wondering if I could grapple or restrain an enemy then bonus action my breath attack on it. Would it cause advantage because the enemy can't move and would it cause any damage to me?  What are the mechanics for this?

1. Dragonborn Breath Weapon is an action, not a bonus action
2. Grappled and Restrained are two different things
3. Grappling a creature just reduces it's speed to zero, no other effect, no advantage
4. Restrained creatures have disadvantage on Dexterity saving throws, which only apply to the following breath weapons:
   1. Black
   2. Blue
   3. Brass
   4. Bronze
   5. Copper
5. You cannot damage yourself with your own breath weapon, you always have to draw it out as a path from you, even if firing straight down

The language under the Dragonborn race mentions specifically that using your Breath Weapon is an Action.  So it cannot be a Bonus Action, or lords no, a Free Action.   As to whether or not you can get advantage, the question is more how can you apply disadvantage on the foe's Saving Throw.  Restraining, Paralyzing, Petrifying or in any other way making the foe make their check at disadvantage or outright fail will work.  There is no language that says that "real world physics/chemistry applies" so you won't hurt yourself doing this trick. The easiest way would be to initiate a grapple and attempt to restrain, then on your next action use Breath Weapon.

I gotta ask though...why?  Your Breath Weapon is weak-sauce and you have dozens of more advantageous actions to use it with.  If you want to be cause it "Looks Cool!" I am 100000000000000000000000000000000000000000% behind you!  If you think it is a mechanically good choice, find another path.

Thank you for the clarification on Grappled and Restrained. I was think of Action Surge and Extra Attack not bonus action.

I was thinking of a How do you want to do this kinda thing.