To make the pastry dough: Combine all-purpose and whole wheat flours and salt in the bowl of a food processor fitted with the metal blade. (See Note for hand mixing instructions). Pulse a few times to combine the dry ingredients. Add the butter pieces and pulse until the mixture resembles pebbles and coarse sand without any pieces of butter larger than a lima bean. Whisk the egg yolk and the ice water, remove the lid of the food processor and drizzle the liquid evenly over the flour-butter mixture. Replace the lid and pulse just until the dough begins to hold together. Do not over mix. You will still see tiny bits of butter in the dough. Turn the dough out onto a large piece of plastic wrap and press into a disk about an inch thick. Wrap well and refrigerate at least one hour or up to one day ahead. (You can also freeze pastry dough, tightly wrapped, for up to 3 months. Thaw in refrigerator before using).
Heat 1 tablespoon olive oil in large skillet over medium heat; add onion, breaking up slices and cook, stirring for 1-2 minutes. Reduce heat to medium-low and cook slowly, stirring often, until onion is soft, sweet, and lightly browned, 20 to 28 minutes. If onion starts to brown too quickly, reduce heat to low and add 1 to 2 teaspoons more olive oil. Reserve. (Caramelized onions can be made up to 3 days ahead and refrigerated, tightly wrapped; warm slightly in a pan over low heat before proceeding).
Preheat oven to 400ºF.
Remove pastry dough from refrigerator and place in center of a 15-inch square piece of lightly floured parchment paper. If the dough is too cold to roll easily, let it stand for about 5 minutes to soften slightly. Roll dough with a rolling pin, starting from the center of dough and rolling toward the outside edges, to a 12-inch circle. If the dough begins to stick to the rolling pin, sprinkle the dough with a little flour. Slide dough circle along with the parchment paper onto a large rimmed baking sheet.
Leaving a 1-1/2-inch border uncovered around the edges, layer the filling as follows onto the pastry: Spread honey mustard evenly onto pastry. Combine the cheeses and sprinkle half onto the mustard, then place the prosciutto in a single layer on the cheese. Add the onion in an even layer and sprinkle with half the herbs. Arrange the sliced pear in overlapping circles and top with the remaining cheese, allowing the pears to peek through the cheese shreds. Sprinkle the remaining herbs over all. Turn over uncovered dough border onto filling and gently press folded areas. Lightly whisk reserved egg white and brush pastry edges; sprinkle pastry and filling with flaky or coarse salt if desired.
Bake in the center of the preheated oven until pastry is golden brown and cheese is melted and lightly browned, about 30 minutes.
Remove galette on baking sheet and set on cooling rack. Serve warm or at room temperature.
Makes 8 to 12 servings.