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The entire sentence reads,
Choose a specific process that would create the most impact the quickest.
In other words, the resulting impact occurs more rapidly and more powerfully than that caused by a different process.
I want to provide a reason why, grammatically, the sentence needs rewording.
While quickest can function as an adverb, "He did it quickest", it's forcing a superlative adjective into that role. What "He did it quickest" is saying is that "He was [the] quickest doing it". It's a colloquialism.
This is why it sounds awkward in your complex sentence: a more complicated sentence needs a more rigorously "correct" form which actually uses a true adverb, most quickly.
It's also A Good Thing to put the adverb closer to the verb it modifies.
Choose a specific process that would most quickly create the most impact.
Changing the order can subtly change the emphasis; in your original the emphasis is on speed, whereas in my example here it's on the impact. However, since speed and impact are both to be maximised, it's a distinction without a difference.
There seems to be a view, expressed in comments, that this is a grammatical structure, in which “quickest” is being used as an adverb.
Although @BillJ is often authoritative, this sounds wrong to me, as it apparently does to @Kate Bunting.
How do I boil water the fastest? Comments would suggest that this is fine. To boil water the fastest, cover the pot.
Surely this deserves examples of this conversion of an adjective to an adverb.
