To prepare crust: Combine all-purpose flour, whole-wheat flour, baking powder, sugar and salt in a food processor; pulse several times. Mix water and oil; sprinkle over the dry ingredients and pulse just until blended. Add olives and pulse to mix. (Alternatively, combine dry ingredients in a large bowl. Make a well in the center and add the water-oil mixture, stirring until well blended. Stir in olives.) Press the dough into a disk; if it seems dry, add a little more water. Wrap in plastic wrap and refrigerate for 30 minutes or longer.

Meanwhile, preheat oven to 400 degrees F. Coat a large baking sheet with cooking spray.

To prepare filling: Combine carrots, parsnips, squash, beet, 1 tablespoon oil, rosemary, salt and pepper in a large bowl; toss to coat. Spread the vegetables on the prepared baking sheet. Cut the tip off the head of garlic. Set on a square of foil, sprinkle with a tablespoon of water and pinch the edges of the foil together. Place the packet on the baking sheet with the vegetables. Roast, stirring the vegetables every 10 minutes, until they are tender and beginning to brown and the garlic is soft, about 35 minutes. (The garlic may take a little longer.)

Transfer the vegetables to a bowl. Unwrap the garlic and let cool slightly. Squeeze the garlic cloves into a small bowl; add the remaining 1 tablespoon oil and mash with a fork. Add the mashed garlic to the roasted vegetables and toss to mix. Add 3/4 cup goat cheese and toss to coat.

To assemble galette: Roll the dough into a rough 14-inch circle about 1/4 inch thick. Coat a baking sheet with cooking spray and place the dough on it. Arrange the roasted vegetables on the dough, leaving a 2-inch border all around. Fold the border up and over the filling to form a rim, pleating as you go. Scatter the remaining 1/4 cup goat cheese over the vegetables. Stir egg and water briskly; brush lightly over the crust.

Bake the galette at 400 degrees F until the crust is golden, 30 to 35 minutes. Let cool for 10 minutes. Serve warm.

Ingredient Notes: Milled from soft wheat, whole-wheat pastry flour contains less gluten than regular whole-wheat flour. It helps ensure a tender result in baked goods while providing the nutritional benefits of whole grains. It is available in natural-foods stores and large supermarkets. Store it in an airtight container in the refrigerator or freezer.

The vegetables should be cut to a uniform size so they cook evenly. Aim for 3/4-inch pieces.

MAKE AHEAD TIP: Store the unbaked crust (Step 1) well wrapped in the refrigerator for up to 2 days.