I stopped BF almost a year ago and I had three white bits on my nipples, two worked their own way out a couple of months ago (it’s milk, looked a bit like cheese like you said) one is still in there but I CBA squeezing it out I’ll come in its own time 😂
It could potentially be an infection. I had one about 4 months into exclusively pumping and it wasn’t too bad at first, pretty similar to how you described but quite quickly got worse. When I eventually went to see my doctor she audibly gasped when I showed her 😂 some antibiotics sorted it out in a week. I would get checked just incase.
Maybe tmi but I'm in the exact same boat. Personally thought it's dryness, are you using nipple balm or any moisturizing?