Ethernet and charge via USB type C I have an HTC 10 with the new USB Type C connector. As USB C is also the primary (or only) connector on some laptop style devices such as the Pixel C  and new MacBook, there are many Ethernet and USB hub device - some with some with power. 

Will this allow me to have a wired network and charge? 

This is essentially the function of the micro USB Ethernet adaptor for Chromecast
It depends completely on the model/brand of the mobile phone - more exactly which drivers for e.g. ethernet chips/adapters that is has integrated in its hardware (since such drivers are not installed in software on mobile phones like this, but they are rather included in the hardware from the start or not).

The following is for example a very common USB-to-ethernet adapter, working for quite many mobile phones (just use a USB-micro-B-to-USB-C wire adapter to be able to plug it into a USB-C mobile phone), but as you can see in the compatibility list a bit down on the page, it depends very much on models and brands of the phones if it will work or not for a specific mobile phone:

https://plugable.com/products/usb2-otge100/

(your specific mobile phone model is not currently in that list, but it lists HTC ONE M9 as working, while listing HTC Desire 612 as not working)

There are also USB 3-based Gigabit ethernet adapters out there (including from the brand in my link above), but beware that these work for a lot fewer mobile phone models, due to the much less widespread inclusion of their drivers in current mobile phone hardware.