In a large bowl, whisk together flours, confectioners' sugar and salt. Cut butter into small pieces. With your fingers, work the butter into the mixture until you have coarse crumbs.
Beat the egg with a fork in a small bowl, then add to the mixture. Mix with your hands until the dough just comes together. Flatten the dough into a disk, wrap in plastic wrap, and chill for 2 hours.
Once chilled, put the dough on a large sheet of plastic wrap, placing another sheet on top. Roll the dough into a 12-inch circle. Lift the top sheet of plastic wrap off the dough, and carefully invert the crust into a tart pan. Lift off the second sheet of plastic wrap and press the dough into the bottom of the pan.
Cut off the excess dough by running a rolling pin across the top of the tart pan (if your dough has any cracks, you can fill them in using the excess dough). Chill in tart pan for 30 minutes to 1 hour.
Preheat oven to 325 degrees F.
Cut out parchment paper into a large circle. Line the tart shell with parchment, and fill with dried beans or pie weights. Bake for 20 minutes. Remove the paper and pie weights and bake for another 5 minutes.
Brush the tart shell all over with egg wash. Bake for 5-7 minutes longer, until golden brown. Remove from oven and let tart shell cool to room temperature (about 30 minutes). Reduce oven temperature to 300 degrees F.
Finely chop chocolate and place in a glass bowl.
On the stove, combine the cream, milk and granulated sugar in a medium saucepan and bring to a boil, stirring frequently to dissolve the sugar. Once just at boiling, remove from heat and pour the hot liquid over the chopped chocolate. Let stand for 30 seconds to begin melting the chocolate. Mix with a spatula until the mixture is smooth.
Let mixture cool to lukewarm, then mix in the egg. (Note: I used what was left of my large egg used for the tart shell egg wash, since I didn't have a small egg).
Pour the filling into the baked tart shell.
Place in oven and bake for 25-30 minutes, until the filling is set. It should look like pudding, and move as one solid mass when the pan is gently shaken. Let the tart cool to room temperature. DO NOT REFRIGERATE.
Once cool, unmold the tart. Arrange strawberries on top of the tart. (If using smaller strawberries, leave them whole. For larger strawberries, cut them into halves or quarters).