As
@MillStream mentioned, Anne Fontaine for fancy-very fancy white shirts. Her business is built around them. I have a lovely dress from them, always great quality, but I'd say on the strongly feminine aesthetic.
If I want a reasonably priced classic RTW white shirt, I go for 100% cotton or linen men's brand like Charles Tyrwhitt (lower-price) or Turnbull & Asser (higher-price). Especially a dress shirt for cufflinks. I realise that not all women can fit into men's styles easily or like the buttons to do up on the 'wrong' way round.
Vogue did a feature last year, but these things are just done for clicks and commission affiliate links. I don't think you have to go for designer. Having said that some of my favourite shirts are
Gucci and
Prada. I'd check out FarFetch or N-A-P, you can see what's on offer across brands with just a word search.