The easiest solution is just to make your own favicon.ico and put in in your ROOT context as a static file to serve.
Otherwise, if you want to tweak the existing id="DefaultHandler" component with your ${jetty.base} take a look at this example ${jetty.base} ...
$ tree -F no-favico-base/
no-favico-base/
├── etc/
│ └── tweak-defaulthandler.xml
├── resources/
│ └── jetty-logging.properties
├── start.d/
│ ├── deploy.ini
│ ├── http.ini
│ └── tweaks.ini
└── webapps/
4 directories, 5 files
It has 2 files that are used to accomplish this.
etc/tweak-defaulthandler.xml
<!DOCTYPE Configure PUBLIC "-//Jetty//Configure//EN"
"http://www.eclipse.org/jetty/configure_10.dtd">
<Configure id="DefaultHandler" class="org.eclipse.jetty.server.handler.DefaultHandler">
<Set name="serveIcon">false</Set>
</Configure>
Which configures the existing id="DefaultHandler" that was previously defined in the ${jetty.home}/etc/jetty.xml.
And an ini to add this xml to the configuration.
This name can be anything you want, I just chose "tweak" as I like to put all of the tweaks I make in this one ini.
start.d/tweak.ini
etc/tweak-defaulthandler.xml
You can see the order of the XML's being executed using --list-config
$ java -jar /opt/jetty/jetty-home-10.0.6/start.jar --list-config
...(snip)...
Jetty Active XMLs:
------------------
${jetty.home}/etc/jetty-bytebufferpool.xml
${jetty.home}/etc/jetty-threadpool.xml
${jetty.home}/etc/jetty.xml
${jetty.home}/etc/jetty-webapp.xml
${jetty.home}/etc/jetty-deploy.xml
${jetty.home}/etc/jetty-http.xml
${jetty.base}/etc/tweak-defaulthandler.xml