You can use the very underestimated Services Menu. Just mark any text, a single letter is sufficient, and control-click to open the context menu. At the bottom you will find the item "Send selection by mail" or "New mail with selection" or something similar.
If you don't see it, open system settings -> keyboard -> shortcuts -> services -> text (or similar, my Mac talks German with me) and enable "send selection by mail". If you don't find "send selection by mail" just add it using the Automator.
Once setup (usually no setup is required, since it is enabled by default) is completed, this is the fastest way to get a New Mail window.