Q.Improve Clay Soil For Desert Plants?
I live in a desert, and want to plant desert plants like thyme and oregano, but the soil where my garden is is an even split of clay and sand, forming concrete. I need to improve the drainage tons, but I don’t want to add tons of organic matter, since they prefer poorer soil. I’ve already added 2in of compost as well, but while it improved the drainage, it also improved water retention tons. While that is normally desirable, it is not in my case.

Certified GKH Gardening Expert
It sounds like you have already done what it takes to amend the soil for the plants. Make sure it is tilled together, thoroughly, so that the organic matter does not sit on top of the hardened layer.
Here are some articles to help you care for the plants, and know their requirements:
https://www.gardeningknowhow.com/edible/herbs/thyme/tips-for-growing-thyme-in-your-garden.htm
https://www.gardeningknowhow.com/edible/herbs/oregano/learn-how-to-grow-oregano.htm