Wildcard dns would work:
*.example.com. 14400 IN A 1.2.3.4
after that just collect webserver logs.
*.example.com. 14400 IN A 1.2.3.4
after that just collect webserver logs.
In case of DNS prefetching, a wildcard record wouldn't be needed, you just need to control the nameservers of the domain and enable query logging.
But I'm not sure how do browsers decide what links to DNS prefetch, maybe it's not even possible for links generated with JS or something like that ... I'm just guessing.