Proof the yeast: Whisk together the warm water and yeast in a small bowl. Let sit for 5 minutes until it foams a bit.
Melt the butter and stir in the vanilla.
In a large bowl, whisk together the flour, sugar, and salt until thoroughly mixed. Add the yeast, butter, egg and egg white. Roughly pull the dough together with a large metal spoon by stirring a few times. Dump the dough (it will be just starting to come together) onto a well-floured work surface. Knead the dough only long enough to form a smooth dough - this took me about 5 minutes.
Transfer the dough to a large greased bowl. Cover loosely with plastic and a cloth and let rise in a warm place free of any drafts for 1 1/2 hours. I have a bread rising feature on my oven, but another effective place is the cavity of a microwave oven. I first heat up a cup of water in the microwave, move the cup to a back corner, and slide the bowl of dough in. Close the door, and you have a perfect, warm, incubator for your yeast to expand.
Stir together the brown sugar, cardamom, and salt in a small bowl.
Butter a 9 x 5 loaf pan. Cut a piece of parchment paper to fit the bottom of the loaf pan, and fit it into the pan. It should adhere to the butter.
Once the dough has risen, turn it out onto a large and well-floured work surface. Roll the dough into a large, thin rectangle, roughly 12x24 inches. Feel free to gently pull at the dough to form the shape you want. This isn't pastry dough. Pour the melted butter over the dough and spread it with your hands so that the entire surface is basted. You can brush it on, if you've got oodles of time on your hands, but I like to do it with my hands. Sprinkle with the brown sugar mixture, reserving 2 tablespoons for sprinkling over the loaf just before baking.
Use a pizza cutter to slice the dough into 6 equal strips along its length, so that each strip is about 24" long. Sprinkle the first strip with 1/5 of the plums - it doesn't have to be exact. Next, lay the next strip on top of the first one. I found it easiest to gently raise one end and then the other, and then slide my arm under the entire strip to move it. It won't break, just stretch. Sprinkle with plums. Continue until all of the plums and dough strips are used, and you have a pile of strips.
Using a sharp knife, cut the stack into slabs that will fit standing up in your bread pan. To get the proper width, I lined up the short end of the bread pan to the edge of the dough, and sliced just a little shorter. Stack the slabs vertically in the pan, like they are papers in a filing cabinet, and continue until all the slabs of dough are in the pan. Don't be afraid to shove in the last couple pieces.
Preheat the oven to 350˚ F. Cover the dough with a towel and let rise in a warm place for about 30-45 minutes. Remove the towel, pour out excess plum juices into a cup or bowl to reserve for yogurt or ice cream, and sprinkle the reserved 2 tablespoons of brown sugar mix over the top of the loaf.
Bake for 45 - 50 minutes until lightly browned and cooked through. You may need to cover the bread with foil after 30 minutes to prevent it from over-browning. Insert a thermometer into the bread - it should reach 200˚ F before removing from the oven.
Cool on a cooling rack for 10 minutes before turning out. The parchment paper should prevent any bread from sticking to the bottom. This bread is best when eaten warm, but try not to eat it all in one sitting :-)