Hello Xth Squad,
You will receive a main mission to hunt down a specific Variant and at that time will be given the Encounter Gauge. It has 6 levels that gradually increases by fighting enemies and increases the levels of the enemies in the area by 1 per level of the Encounter Gauge. Simply reach level 6 on the Encounter Gauge and fight 20 battles.
They do not have to all be done in a row but they do have to be completely separate battles so those battles where more enemies show up after defeating what appeared to be all enemies will only count as 1 battle no matter how many times more keep showing up.
If there is anything I can do to make this solution better please don't hesitate to let me know.