You can make multiple attacks with the Plasma Blast feature
The effect of using the plasma blast is pretty simple, because it doesn't grant you a brand new attack type or anything like that, it just modifies your existing unarmed strikes:
Plasma Blast: You condense your body’s qi into a beam of superheated plasma, allowing you to make unarmed attacks at range, with a range increment of 30 feet. All your unarmed attacks deal electricity and fire damage and gain the stunned critical hit effect.
There is no special rule that you can only do this once a turn or anything like this. You just now have unarmed strikes that work at range and do energy damage, and you can use them as many times as you'd normally be able to do so; which is indeed twice, if you make a full attack, or 3 times with Soldier's Onslaught. The only reason you wouldn't be able to do so is if you activate plasma blast using the enhanced mode from the Bountiful Qi feature:
Plasma Blast: Choose one of the following weapon special properties: automatic, blast, explode, or line. Each time you attack with your plasma blast, you can make the attack with the chosen weapon special property and the unwieldy weapon special property.
When you use your plasma blast in this enhanced way it has the unwieldy property, the only effect of which is that you can't use such a weapon more than once a turn (or to make attacks of opportunity). So it stands to reason that since this restriction is explicitly made for the enhanced version of plasma blast, there is no such restriction when you use it in the normal way.
So, that established - if you are a 20th level soldier with the Qi Adept fighting style and the Aesthetic Warrior alternate class feature, then yes, you have unarmed strikes/plasma blasts that do 7d6+30 damage and can attack up to three times using Soldier's Onslaught. (Compare to a 20th level soldier simply using a zenith-quality laser rifle, who can do 11d6+20 damage per attack but at much greater effective range.)
I'm not sure you should be getting any dexterity bonuses to damage, though; at best you'd get to add strength to hit and damage, if you're treating the attack as if it were a thrown melee weapon - if it's actually a standard ranged attack, you'd just add dexterity to hit and have no extra damage.