I don't know if there's an MQTT bridge yet. Generally, you run some kind of border router and connect through that. While the work doesn't seem complete yet, there's a guide to turn a Home Assistant install into a Thread border router if you have the necessary hardware: https://www.home-assistant.io/integrations/thread/#turning-h...
You can probably plumb Home Assistant into your MQTT server from there.
That is my sticking point - every border router I can find has a bunch of other things I don't want. I don't want my lights available from the internet, I just want to turn on the shed outside light from inside the house when I have guests. (since they are likely to park by the shed and walk to the house)
That is I don't want google/amazon/samsung/apple to control my house. Most border routers are also connect to our smart home system. (there are exceptions but it isn't clear if they are better)
While not a proper product, you can buy a small ESP32C6 devboard for up-to-5 USD/EUR and flash an example from esp-idf. This is paired with some software that runs on posix systems (think your HA host) that together form the necessary commissioning and border routing functionality. Its ends up being a relatively simple device that just takes IP packets off the radio and puts then somewhere else, so I've no doubt somebody will shortly make one (I've been working on one such for myself as an experiment.)
For what you're looking to do in principle you don't really need any of this after the initial commissioning. So long as the radio waves can reach the devices they will be able to talk to each other.
Just checked again, those ideas have started to mature to something that is possible now (vs 6 months ago when I last looked). I have kids and many other things in my life, so I'm don't have much time to work on projects like this. That might be what I end up doing, for now I just have a light that hasn't worked for months because I don't want to figure this out (the manual switch is broken. Since the switch is both rarely used and one I'd want remotely controlled anyway I've been hesitating)
SMLight also do their PoE sticks that can be flashed to either talk Zigbee or Thread (but you'll need a border router, such as the OpenThread border router).
Their latest one has two radios so you can do both Zigbee and Thread from a single device.
I've found however, that Thread prefers several border routers around my house to operate well.
