I have this issue with the latest Cisco Anyconnect & Big Sur update as of 2021-07-19. For me, the sudo killall -HUP mDNSResponder is inconsistent and works only for a few minutes. A better solution for me is to use Google (or cloudflare DNS) servers. Go to Settings -> Network -> Advanced -> DNS tab Click '+', then add server 8.8.8.8 Delete any other DNS servers in this list by selecting them and pressing '-' (minus) Click OK, then Apply, then restart. I haven't encountered any problems with this approach so far.

See my answer here: https://developer.apple.com/forums/thread/667347?answerId=682255022#682255022