My whole background is related to enterprise applications but I've been thinking of switching my career path from enterprise/business to something that I feel is more challenging and interesting. One of the main fields I've checked is game programming and more specifically engine or graphics programming. One of the requirements I've set is that the new domain requires at least some math/computer science knowledge and have interesting problems to research and solve (hopefully even read some white papers). Graphics programming especially seems to have lot's of that in addition to being interesting in general. I've already started studying the related mathematical background via this book and I believe my general experience in developing and releasing quality software (5+ years) will quickly translate, with the right mentorship, putting me at a level of usefulness of a mid-level game programmer at least.
How should I go about proving this to potential employers? I don't feel like creating small sample games really amounts to anything. Familiarizing myself with an engine like Unity or Unreal seems like a good approach but I'm not sure how to go about it in an engine focused way. I'd potentially like to avoid Gameplay development which is why creating games seems redundant.
I don't want to enter subfields such as server programming or any field that is web-appy but under the umbrella of a game company. I'm also looking to transition relatively quickly which is why I don't want to invest in a game company as a web app developer only to hope to be put into game domain later.
TL;DR: I'm looking for ways to enter the "sciency" side of game programming (graphics/engine programming) with a background in enterprise web apps.