In a medium saucepan, combine water, salt and tapioca. Cook over medium heat for 15 minutes.
Add the brown sugar to the tapioca, lower the heat and cook for 5 more minutes. There will still be a bit of white left in the tapioca, if it is more than halfway white cook a bit longer.
Whisk together the egg and milk in a measuring cup. When the tapioca mixture is mostly cooked, scoop out about ⅓ cup and whisk it into the milk and egg. Then whisk the tempered egg mixture into the rest of the tapioca.
Cook over low heat for about 10 more minutes. While it is cooking, in a small saucepan melt the butter and continue cooking until it is light brown.
Combine browned butter and vanilla with the tapioca. Pour into one 1 quart serving dish or 4-6 individual serving dishes. Let cool to room temperature and serve.
Notes
WARNING: Not all small pearl tapioca is the same. If the recipe on your package calls for less tapioca per 2 ½ cups of water use the amount on the package to start and add more if it seems soupy. I have found a stark difference between brands with some needing half as much.