Maybe I'm picky here, but for curiosity sake I wonder if it's really worth using lambdas here when the difference is at most just one line of code. Without passing a lambda it would be almost the same from the reading complexity perspective.

    final String uri = request.getUri();
    final URI parsed = URIUtils.safeParse(uri);
    if (parsed != null) {
          if ((uri.getHost() == null || uri.getHost().equals(""))
              && request.headers().contains(HOST)) {
            return URIDataAdapterBase.fromURI(
                "http://" + request.headers().get(HOST) + request.getUri(),
                URIDefaultDataAdapter::new);
          }
          return new URIDefaultDataAdapter(uri);
    }
    return new UnparseableURIDataAdapter(uri);

It would open opportunity to remove the mapper parameter completely and just use URIDefaultDataAdapter's constructor within URIDataAdapterBase.fromURI for simple use-cases.

Nice to have it covered with a test.

As I mentioned before, I'm not completely sure if the mapper parameter is worth it. With or without it it's the same amount of boilerplate for the custom use-cases, and is more lightweight for simple cases.

I agree with you. I leverage the mapping in the netty decorator, where the parsed uri is used, eventually, with host coming from the http headers to fix the url. So having a mapping function saves me some if and some other boilerplate to parse the URI. I may think about a better way to do it without having a mapper fuction and improve in a next PR

Nice