Put the water in a large bowl (or the bowl of your stand mixer) and sprinkle with the yeast and a pinch of the sugar. Let stand for 5 minutes, or until it dissolves and gets foamy.
Add the flour, the rest of the sugar, the milk, egg, butter and salt and stir by hand or with the dough hook attachment of your stand mixer until you have a soft dough. Knead for about 5 minutes, or until it’s smooth and elastic—it should still be quite tacky, but will swell and smooth out as it sits. Place the dough back in the bowl, cover with a tea towel and let rise in a warm place for an hour or so, until doubled in size.
Roll the dough out into a rectangle that’s about 12x16-inches, spread or brush with very soft or melted butter, sprinkle with brown sugar (once scattered overtop, I spread it out evenly with my hand) and sprinkle it generously with cinnamon.
Cut the rectangle lengthwise into 4 strips. Roll up one of the strips, then move it over to the end of another strip of dough, and pinch the ends of the dough together. (It doesn’t need to be perfect—they will bake together inside the roll.) Continue rolling up that strip, and then carefully continue with the remaining two strips, making one giant roll. I use a bench scraper to gently ease up the dough, and then transfer the whole roll into a buttered or parchment-lined 9-inch square or round baking dish, deep cake pan or ovenproof skillet. Scatter any brown sugar that has fallen out onto the counter over the dough. Cover with a tea towel and let it rise for an hour.
When you’re ready to bake, preheat the oven to 350F. Bake the mega roll for 50-60 minutes, or until deep golden and puffed. Whisk together the melted butter, icing sugar, milk and vanilla, adding a bit more liquid or sugar as needed so that it has the consistency of heavy cream, and drizzle it over with a fork while the bun is still warm. Serves 4-6 (or 1).