If you have left-over beetroot, then I’d want you to try the following recipe, inspired by the Roasted Pepper (Capsicum) and Feta sauce called Htipiti. At this stage, I haven’t tried it myself yet, but I bet it works.
It’s easy. Put the boiled beetroot in a food processor and puree until a chunky sauce. Add the feta, keep processing. Add any leftover dressing, a little olive oil, garlic oil, or chilli oil (or olive,garlic and chilli oil if you have made some)
Taste, and adjust the seasoning until you like it. A little extra honey may appeal to you. Salt is unlikely to be necessary as feta is a salty cheese anyway.
I’d also try this with carrot, maybe with coriander (fresh or dried) or cumin seeds.
Update: It does work. Made with chile oil, and thinned with a little yoghurt, it works as a pasta sauce too.