I've dropped the framerate and resolution to their lowest settings, and set the graphics to lowest quality, but Hat in Time continually stutters in both pictures and music.

My rig is:

Windows 10 Pro, 64-bit
Intel Core i5 4690K running at 3.5GHz
8GB RAM
GeForce GTX 750Ti

So, what am I missing?

I am having the same issue. The game used to work just fine, but the current release stutters. I can run games that are more graphic intensive and they work fine.

Windows 10 Home, 64-bit
AMD FX 8350 running at 4.0Ghz
32GB Ram
Radeon RX 570 (8GB)

The new update Nyakuza Metro v1.01 seems to have alleviated some of the stuttering. The stuttering subsides a bit after playing the game a bit, but does not go away entirely.

The 1.03 patch fixed the stutter issue.