Steven Soderbergh, Reed Morano, Nicolas Roeg, and Shane Carruth come to mind. David Lynch did once, and of course Alfonso Cuarón recently.
I do my own cinematography when I direct short films, but they're very modest in scope. The thing I've heard in interviews with those directors is to rely heavily on your other crew, especially your gaffer (Paul Thomas Anderson simply dropped the cinematography credit on Phantom Thread completely because he saw it as a group effort between himself and his crew). Do a pass at lighting, and while your crew works, you work with your actors.
I think some styles are better suited for it than others. Reed Morano, for example, tended to have the camera on her shoulder and become a very active participant in the scene with her actors. Soderbergh, at least more recently, has been trying to minimize his equipment needs so he can begin shooting as fast as possible, even an iphone and just one or two LED panels. Personally, I almost always shoot on a tripod with no camera movement whatsoever with very few close-ups. This allows me to light a space, compose a frame and block a scene, then focus almost entirely on the actors when the camera is rolling. It's a very specific, formal look, but it suits my narratives. If I had more elaborate camera movement, there's no way I'd be able to handle both jobs at once.
It's a fun challenge, but you're definitely limiting your options by doing both. I know I'm a better cinematography when I work with other directors, but because I do such a specific thing with my own films, it works for me. I certainly wouldn't try it for the first time on a project you can't afford to mess up.