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For the filling: Preheat the oven to 400°F. Toss the diced sweet potatoes in the olive oil and spread on a baking sheet. Sprinkle with salt and roast until just soft enough to easily pierce with a knife, about 20 minutes. Remove from the oven, let cool completely, and place in a medium bowl with the red onion.
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To make the sauce: Combine all of the ingredients in a blender; blend on high until smooth. Pour the sauce over the roasted sweet potatoes and red onion, stirring until well coated. Set aside.
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To make the dough: Combine the water, milk, and olive oil in the bowl of a stand mixer fitted with a paddle. Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the flour, yeast, and salt. Mix on low speed until a smooth dough forms.
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Scrape the bowl and switch to a dough hook; knead at medium-low speed for 8 minutes. If needed, turn the dough out of the bowl and finish kneading by hand on a lightly floured surface. Transfer the dough to a lightly oiled bowl, cover, and let rise at room temperature for 1 hour.
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To shape: Lightly flour your work surface. Roll the pide dough to a long oval about 17" long and 6" wide. Place on a parchment-lined baking sheet.
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Add the crumbled feta to the sweet potato mixture. Spread an even layer of the filling down the center of the dough, leaving a 2" border uncovered along each side. Fold the long sides just over the filling, leaving the center open.
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Pinch the ends together to seal. Cover with greased plastic and let rise at room temperature for 30 minutes.
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Preheat the oven to 400°F.
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Brush the pide with the egg wash and bake for 25 to 30 minutes, until golden brown. Remove from the oven and let cool completely. Sprinkle the pumpkin and pomegranate seeds over the top before serving.
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Store, well wrapped, in the refrigerator for up to five days.