First of two encounters with Brooke Augustine takes place while you're exploring the prison on the island during the fifteenth mission of the game (Quid pro Quo). Encounter begins soon after reaching the area and watching the cut scene with Reggie.
Encounter with Brooke Augustine takes place on the big arena where you can move freely. As like in previous boss fights, try to keep Augustine at bay and don't get closer for melee attacks. Important is also to keep being mobile and/or hiding behind covers while Augustine throws things at you.
The best is to use the strongest Smoke Attack, which is Cinder Missile (R1), to attack Augustine. Notice that each hit destroy part of her cover.
Important thing about attacking your opponent is that you can't stop it for a longer period of time, as she will regenerate. So you should have a lot of heavy ammo (try to increase its limit with upgrades) and replenish it regularly. In this area you can find a lot of places to regenerate Smoke, but use them only when Augustine finishes her attack. Otherwise, your action may be interrupted (here an upgrade which decreases the time of draining the smoke may be handy).
When you destroy Augustine's cover, make some more attacks to hurt her directly. Fire until woman lands on the arena. In such a situation, you need to run up to Augustine and press the Triangle (if you play as the hero), or the square (if you are playing as the villain) to land a powerful kick (the above screenshot).
Further part of the first phase doesn't differ too much. You still have to destroy Augustine's shield and melee attack her only when she falls on the ground. After several sequences like that, game will let you to fire a special attack. You don't have to make anything special as this attack always hits. It means also that the second part of the encounter begins
Don't think that you'll be able to resume attacking Augustine at once, as she'll leave arena and send a group of soldiers at Delsin. Don't underestimate those enemies. Many of them have superpowers (and those should be dealt with as first ones) and many covers are already destroyed here. Don't stop for too long anywhere and don't let enemies get close to Delsin. Use the better attacks (rockets also) always when it's a chance to attack a stronger enemy or a group of them.
Augustine rejoins the battle after you eliminate the main group of DUP soldiers. Notice that there are several new soldiers at the arena. You should get rid of them ASAP, because there'll be no more enemies. Augustine will move around the battlefield so pay even more attention to keep away from her.
Second phase of the encounter is very similar to the first one, so you again have to make strong distance attacks to destroy her shield. A difficulty may lay in a fact that you lose all places to regenerate Smoke, but, luckily, it doesn't weaken the main character. Why? Because now you can look for Vide regeneration points. Use the strongest power (Bloodthirsty Blades) which is as effective as rockets.
Continue attacking, using Smoke or Video whichever suits you. As in the first part of the encounter, after which series of hit, you have to kick Augustine when she lies on the ground. After several sequences like that, you may perform a special attack. Making it, ends this encounter.