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To make the poolish: In a large bowl, stir together the flour, water, and yeast. Set aside, covered, to rest at room temperature for 6 to 12 hours; overnight is fine.
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To make the dough: Add the flour and water to the poolish, mix well, and let rest for 20 minutes.
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Add the remaining dough ingredients, mixing and kneading briefly to form a semi-smooth dough. The dough should be slightly sticky and soft, and may still have a rough surface.
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Cover and let the dough rise for 45 minutes.
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Gently fold the edges of the dough into the middle, turn it over, and let it rise for an additional 45 minutes. Prepare the filling while the dough is rising for the second time.
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To prepare the filling: Clean and remove any large or tough stems from the spinach. Set aside.
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Heat a large skillet until hot, add the oil and garlic, and immediately add the mushrooms. Stir with a heatproof spatula; the mushrooms will begin to release water.
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After about 2 minutes add the spinach, salt, and pepper. Cook just enough to wilt the spinach. Add the feta and olives. Remove the pan from the stove and set the filling aside until you’re ready to use it.
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To assemble the pizza: Place the dough on a lightly floured surface and divide it in half. Roll, then stretch one piece of the dough into a 14” circle.
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Place the dough on a baking sheet or round pizza pan that's at least 14" in diameter.
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Top with the filling, leaving a 1/2" border around the edge of the dough.
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Roll the second piece of dough into a 14" circle. Place it atop the filling, gently pressing it down.
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Pull the edges of the bottom dough up and over the edges of the top dough and press together to seal. Cut several small holes in the top crust to allow steam to escape while baking.
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Let the pizza rise for 20 to 30 minutes while you preheat the oven to 500°F.
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Bake the pizza for 30 to 35 minutes, until it’s golden brown all over.
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While the pizza is still hot, brush the top crust with olive oil and sprinkle with flaky sea salt, if desired. This will add a nice shine to the crust and additional flavor.
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Let the pizza cool for about 5 minutes before cutting so the filling has time to settle. Cut into slices and serve warm.
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Storage information: Store leftover pizza, well wrapped, for several days in the refrigerator. To reheat, place slices on a parchment- or foil-lined baking sheet and warm in a preheated 350°F oven for about 10 minutes, or until heated through. For a less crispy version, wrap in foil and heat as directed.
