Make the tart dough: In the bowl of a stand mixer, combine the all-purpose flour, whole wheat flour, and kosher salt. Turn on the stand mixer to low with the paddle attachment, and slowly add the unsalted butter, cut into small chunks. Mix until the dough is crumbly with small butter chunks. (Alternately, mix the butter into the dough using a pastry cutter.) Slowly add the ice water into the stand mixer until the dough just comes together. If necessary, add another 1/2 tablespoon of water to bring it together. Remove the dough from the bowl and knead it once or twice until the dough is smooth. Shape it into a rough rectangle and wrap in plastic wrap. If desired, chill in refrigerator for 15 minutes. (If made the day before and refrigerate overnight, leave the dough out for about 30 minutes until soft enough to roll out.)
Make the ricotta filling: In a bowl, stir together ricotta cheese, lemon zest, 1/4 teaspoon kosher salt, and several grinds black pepper. Roll out the galette dough to a large, thin rectangle, then transfer to a parchment-lined baking sheet. Spread the ricotta filling on top in a thin layer to about 1/2-inch from the edge of the dough. Brush the exposed dough with the melted butter. Bake 30 minutes until golden brown.
Make the blackberry onion sauce: Thinly slice the onion and measure out 1/2 cup. Heat 1 tablespoon of olive oil in a small skillet over medium heat. Add the onions and cook about 5 minutes until translucent, stirring occasionally. Add the balsamic vinegar, reduce the heat to low where it is lightly bubbling, and continue to cook for 10 minutes until the vinegar is cooked out. Meanwhile, drain and measure the blackberries and measure the thyme leaves.
To the skillet, add 2/3 cup water, blackberries, thyme leaves, and a scant 1/4 teaspoon kosher salt (start conservative). Simmer 3 minutes. Remove a few spoonfuls of the juice in the skillet into a small bowl, then stir together with 1 teaspoon cornstarch until smooth. Pour the cornstarch mixture back into the skillet and stir until the sauce becomes thickened. Remove from the heat and stir in 1 teaspoon lemon juice. Taste, and adjust as necessary (the salt and lemon should pop the flavors, but not be salty).
When the crust and ricotta are done, spread them with berries and top with fresh thyme leaves. Slice into pieces and serve.