Probe for Brave-owned domains instead of Google ones when user selects custom DoH provider

Description

As discussed in https://github.com/brave/brave-browser/issues/38058, when the user chooses a custom DoH provider, Brave asks the new DoH provider to resolve gstatic.com and google.com. Importantly, the connection is not actually made to these domains, it's simply a test query that tells the DoH provider that some Chromium-based browser has selected it to be its custom DoH provider. This is not a privacy leak, but it might be worth patching out just for appearance sake ("why is Brave doing a DNS query for google domains?")

See: https://github.com/GrapheneOS/Vanadium/blob/main/patches/0064-use-grapheneos.network-when-verifying-DNS-over-HTTPS.patch

Steps to reproduce

Follow the steps on https://github.com/brave/brave-browser/issues/38058#issuecomment-2118199829

Actual result

DoH query for google and gstatic

Expected result

DoH query for a Brave-owned domain

Reproduces how often

Easily reproduced

Desktop Brave version (brave://version info)

All

Android device

All

Channel information

• release (stable)
• beta
• nightly

Reproducibility

• with Brave Shields disabled
• with Brave Rewards disabled
• in the latest version of Chrome

Miscellaneous information

No response