I had nearly the same issue of Laravel Valet not working after installing via Homebrew on macOS 10.13 "High Sierra".
The problem I encountered was that DNSMasq would respond to queries, but Nginx would not handle the responses on port 80.
Attempting to connect to the site in a Terminal session would result in the following error:
$ curl -IL http://example.test/
curl: (7) Failed to connect to example.test port 80: Connection refused
I had recently upgraded from macOS 10.2 "Sierra" to 10.13 "High Sierra", so I suspected my Homebrew environment and configuration that was brought by the Migration Assistant could be at fault.
While following the troubleshooting advice here and elsewhere, what eventually solved my problem was to completely uninstall and remove Valet along with its dependencies (PHP, Nginx, DNSMasq, etc):
$ rm -rf ~/.valet
$ brew unlink nginx && brew remove nginx
$ brew unlink php56 && brew remove php56 && brew uninstall --ignore-dependencies php56
$ brew unlink php72 && brew remove php72
$ brew unlink dnsmasq && brew remove dnsmasq
With a clean slate, I was able to successfully install Valet:
$ brew update
$ brew install homebrew/core/php
$ composer global require laravel/valet
$ valet install
I then setup a new development domain:
$ valet domain test
$ valet park ~/Sites
$ mkdir ~/Sites/example && cd "$_"
$ valet link
$ echo "Hello, world" > ~/Sites/example/index.html
With Valet installed, I verified everything was working:
$ valet --version
$ sudo nginx -t
$ ping -c 4 example.test
$ curl -IL http://example.test/
$ valet open example
After doing all of this, I had a 100% successful working Laravel Valet environment.
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