1) In a stand mixer or bread machine (programmed for dough), mix together all of the ingredients except the ham to form a smooth, shiny dough. Don't worry; what starts out as a sticky mess becomes beautifully satiny as it kneads. This dough takes longer than most to develop, so be prepared to let the dough knead for up to 15 to 20 minutes in a stand mixer. Also, we don't recommend trying to knead it by hand. Once the dough is fully kneaded, knead in the ham. If you're using a bread machine, add the ham about 5 minutes before the end of the kneading cycle. Let the dough remain in the machine for an additional 60 minutes, once it's completed its cycle; then refrigerate as directed below.
2) Form the dough into a ball (it'll be very soft), place it in a greased bowl, cover the bowl, and it let rise for about 2 hours at cool room temperature; it won't seem to change much, but don't worry. Refrigerate the dough for several hours, or overnight. This will allow fermentation to slowly continue; and will chill the butter, making the dough easier to shape.
3) Remove the dough from the fridge, and place it on a lightly floured or lightly greased work surface. It'll feel very stiff and clay-like. Shape it into a long (28") log. It's best to work as quickly as possible; the warmer the dough gets, the stickier it is, the more challenging it is to work with.
4) Lightly grease a 9" round cake pan. Coil the log into the pan in a spiral.
5) Cover the pan, and let the shaped brioche rise for 2 to 3 hours, until it's come to room temperature and nearly reached the edges of the pan. Toward the end of the rising time, preheat the oven to 400°F.
6) Brush the brioche with the reserved egg white, which you've mixed with 1 tablespoon cold water. This will give it a shiny crust.
7) Bake the brioche for 20 minutes. Reduce the oven temperature to 350°F; gently tent the loaf with foil, and bake for an additional 35 to 40 minutes, until the loaf is golden brown (peek under the foil), and an instant-read thermometer inserted into the center registers at least 200°F.
8) Remove the brioche from the oven, and very carefully slip it out of the pan onto a rack to cool.
9) Serve warm, or at room temperature; refrigerate any leftovers, tightly wrapped. Reheat before serving; a toaster or toaster oven works well here.
Yield: one 9" round loaf.