Put the flours, salt, and butter in the bowl and massage them together with your clean fingers until the mixture looks like little round lumps. Be patient; this will take a little while to get the hang of and to do.
Add the water, 1 tablespoon at a time, massage some more, and form into a ball. Flatten the dough into a large disc, cover with plastic wrap, and refrigerate 1 hour.
Put the apple slices, yogurt, cinnamon, and vanilla in the bowl you just used to make the dough, and mix well.
Take the disc out of the refrigerator and put it on the baking sheet. Using the rolling pin, roll the dough into a 15-inch circle (it doesn’t have to be exact). Sprinkle 1 tablespoon of flour in the center, leaving a 2-inch border all around (this means: don’t sprinkle the flour on the outermost 2 inches). Spread the apple mixture evenly over the flour, arranging the slices so they overlap.
Fold up the dough border to partially cover the fruit and pinch and seal the corners where possible.
Once the oven temperature has reached 400 degrees, carefully put the baking sheet in the oven and bake until the crust is lightly browned and the fruit is bubbling, 35-50 minutes. Set aside to cool a bit and serve warm or at room temperature.