Preheat oven to 350 degrees F. In a medium saucepan melt butter over low heat. Remove from heat, add both sugars, and stir to combine. Stir in the flour and set aside to cool for 15 minutes (the dough will still be a little warm). Press the dough evenly onto bottom and sides of a 9x2-inch tart pan. It is important to fit the dough evenly into every nook and cranny of the pan, especially the scalloped edges. Trim excess dough with a knife.
Place the tart shell on a sturdy cookie sheet so it will be easy to move in and out of the oven. Line the shell with aluminum foil and add pie weights or dried beans to keep the sides of the tart from buckling. Bake for 30 minutes. Take the tart out of the oven; remove foil and weights or beans. Using a pastry brush, lightly coat the shell all over with a thin layer of egg white. Return to oven and continue to bake for another 8 minutes until the shell is cooked through and a light golden color, but not browned. (The tart will be cooked again with the filling; you don't want it to burn on the second bake.) Set aside to cool while making the filling. Reduce the oven temperature to 325 degrees F.
To make the filling:
Combine the heavy cream and milk in a saucepan and heat over medium-low heat until mixture just simmers around the edges. Remove pan from heat, add chopped chocolate, and stir until smooth. Add sugar and salt; whisk until well incorporated. Let sit for 5 minutes to cool slightly. Add eggs, and whisk until completely blended. Pour filling into cool tart shell, and bake for 15 to 20 minutes until the filling is just set and surface is glossy. If bubbles or cracks form on the surface, remove tart from oven-its beginning to overbake. Cool on wire rack. Serve warm or at room temperature. Makes 8 to 10 servings.