First, I'm running macOS High Sierra, with Homebrew's PHP 7.1
I've also followed the Valet documentation to the letter, as well as trying what was on these asks: Valet (Laravel): DNS address can not be found and Laravel valet It works aaaaaand, despite all of that, when I navigate to project.dev, I just get "It works!" I was having issues pinging, but that turned out to be Stealth Mode in the macOS Firewall.
I have Laravel set up in this directory, so I SHOULD be getting the default splash page, right?
Also, in case it matters, here's the line I used to install PHP 7.1 itself: brew install php71 --with-httpd --with-pear --with-postgresql --with-thread-safety --with-homebrew-curl
You should check your settings related to firewall. Make sure you are allowing pings and ICMP.
To check it on a Mac :
System Preference > Security & Privacy > Firewall > Firewall Options
This sounds like the Apache that comes pre-installed with macos is running in the port 80. Before starting valet, make sure that Apache is stopped.
You can these commands: sudo apachectl stop or sudo apachectl -k stop to stop Apache, and after that, then you can proceed with your valet process.
Hope it helps.