Allow ESPHome devices to act as DNS Server

**Describe the problem you have/What new integration you would like**

I have a Huawei AX3 Pro router. It's a great piece of hardware, that unsurprisingly lacks a proper software. OpenWRT is likely to never be available to it.

I use AdGuard Home as a HA add-on, acting as a DNS server, and I have a problem:

When my HA instance goes down, the internet connection of the whole home goes down as well. Like for example, when I'm restarting HA.

But another situation is when I have a power outage, which is frequent where I live. The UPS which holds my HA instance can hold it up for like 15 minutes, but the UPS which holds my Huawei router can hold it up for like 45 minutes. In other words, in a power outage I only get internet connection for the firsts 15 minutes.

I can of course manually go in the Huawei web UI and change it to something like 1.1.1.1 to restore connectivity, but doing that manually every time is cumbersome.

**Please describe your use case for this integration and alternatives you've tried:**

If I could make an ESPHome act as a DNS Server, I'd put it as the DNS address of my router. Inside of ESPHome, it would redirect requests to my AdGuard Home add-on if the HA host is up, or otherwise to 1.1.1.1.

**Additional context**

https://github.com/devyte/ESPAsyncDNSServer
https://github.com/zhouhan0126/DNSServer---esp32

The request names no ESPHome files, tests, or entry points. Start by reviewing the linked ESPAsyncDNSServer and DNSServer projects, then determine how the requested DNS behavior fits ESPHome; done should include forwarding to AdGuard Home when available and falling back to 1.1.1.1.

Written by the indexing model from the issue text.