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For the crust:
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Combine whole wheat flour, all purpose flour, baking powder, sugar and salt in a food processor; pulse a few times.
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Mix water and oil; sprinkle over the dry ingredients and pulse just until blended.
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Add olives and pulse to mix.
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(Alternatively, combine dry ingredients in a large bowl.
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Make a well in the center and add the water-oil mixture, stirring until well blended.
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Stir in olives.)
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Press the dough into a disk; if it seems dry, add a little more water.
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Wrap in plastic wrap and refrigerate for 1 hour or longer.
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Meanwhile, preheat oven to 400F (200C).
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Coat a large baking sheet with cooking spray.
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For the filling:
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Combine all the roots vegetables, 1 tablespoon oil, rosemary, salt and pepper in a large bowl; toss to coat.
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Spread the vegetables on the prepared baking sheet.
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Cut the tip off the head of garlic.
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Set on a square of foil, sprinkle with a little bit oil and pinch the edges of the foil together.
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Place the packet on the baking sheet with the vegetables.
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Roast, stirring the vegetables every 10 minutes, until they are tender and beginning to brown and the garlic is soft, about 35 minutes.
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(The garlic may take about 8 minutes more.)
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Transfer the vegetables to a bowl.
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Unfold the garlic and let cool for a while.
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Squeeze the garlic cloves into a small bowl; add the remaining 1 tablespoon oil and mash with a fork.
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Mix well the mashed garlic to the roasted vegetables.
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Add 3/4 cup goat cheese and toss to coat.
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To assemble galette:
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Roll the dough into a 1/4 inch thick and 14-inch circle rough .
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Place the dough on a baking sheet with cooking spray.
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Arrange the roasted vegetables on the dough, leaving a 2-inch border all around.
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Fold the border up and over the filling to form a rim, pleating as you go.
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Sprinkle the remaining 1/4 cup goat cheese over the vegetables.
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Stir egg and water briskly; brush lightly over the crust.
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Bake the galette at 400F (200C) until the crust is golden, 30 to 35 minutes.
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Let cool for 12 minutes.
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Serve warm.