A double-knitted toe transmogrifying into brioche which works its fairly plain way up to the cuff with a few cable twists, increases and decreases.
This is a mostly charted pattern, using nonstandard brioche symbols (because my design tool does not come with them automatically and I have yet to make my own), and is not test-knitted beyond the original sock. It assumes knowledge of basic double-knitting and brioche, plus some intermediate brioche techniques (cabling).

You will need two sets of sock needles for the toe (or use your own favourite).