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To make the dough: In the bowl of a stand mixer fitted with the dough hook, combine the flatbread mix, water, eggs, oil, and yeast. Knead the dough for 5 to 6 minutes on low speed until the dough is soft, smooth, and elastic. (See “tips” below for details on mixing by hand.)
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Cover the bowl and let the dough rise for 30 minutes.
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To make the filling: Fill a large pot 3/4 full with water and salt to taste. Heat over high while you peel and quarter the potatoes. Once boiling, cook the potatoes in the salted water until tender, about 10 to 15 minutes. Drain in a colander and let the potatoes cool for 5 to 10 minutes.
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In a large skillet, sauté the onion in the vegetable oil over medium-low heat until tender and starting to brown, 7 to 9 minutes.
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Once the potatoes have cooled slightly (they should still be warm but not hot), use your fingers to crumble the potatoes into a large mixing bowl. They'll look like coarse, fresh breadcrumbs. Add the sautéed onions, along with the peas, salt, za'atar packet, and red pepper flakes; stir together. Add the egg, feta, and mint; stir to combine all the filling ingredients. Season with salt to taste.
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To assemble the flatbread pockets: Lightly grease 2 baking sheets or line with parchment.
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Turn the dough out onto a clean work surface and roll or stretch the dough into a rough 17" x 22" rectangle, working to keep the thickness as uniform as possible. Trim the dough to make a neat 15" x 20" rectangle. Use a pizza wheel or sharp knife to cut the rectangle into twelve 5" squares (a 3 x 4 pattern works well here). Once cut, leave the dough on the work surface; this will make assembly much easier. (See "baker's tips," below for details on what to do with the scraps.)
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Place about 1/3 cup (roughly 70g or a rounded scone and muffin scoop) of filling in the center of each of the dough squares, then gently flatten to about 1/2" thick. Starting at the edges of the square. Bring the corners together in the center and pinch each seam to seal in an “X”.
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Gently transfer the pockets to the prepared baking sheets, placing them seam-side down. Cover and let the pockets rise for 30 minutes.
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To bake the flatbread pockets: Towards the end of the rising time, preheat the oven to 375°F with racks in the upper and lower thirds.
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After the 30-minute rise, uncover the pockets, brush with the egg wash, and prick the centers with the tines of a fork.
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Bake the flatbread pockets for 15 minutes, then rotate the pans and bake for an additional 10 to 12 minutes until golden brown. The internal temperature of the filling should be over 200°F when measured with a digital thermometer.
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Remove the baked flatbread pockets from the oven and allow them to cool directly on the baking sheets. Serve warm or at room temperature, alongside your favorite condiments.
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Storage information: Store leftover flatbread pockets, covered, in the refrigerator for several days; freeze for longer storage. To reheat, wrap in foil and place in a preheated 350°F oven for 20 to 25 minutes, or until internal temperature is at least 140°F when measured with a digital thermometer.
