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To make the filling: Fill a 2-quart saucepan with water 2” deep; salt the water generously and bring to a simmer. (Salting the water helps the eggs cook more evenly, as well as helping to prevent any leakage if they crack.) Add the eggs and simmer for 6 minutes.
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When the time is up, immediately rinse the eggs in cool water. Gently crack the shells all over and place the eggs in a bowl of cool water. Carefully peel them, rinse to remove any shell fragments, and refrigerate until needed.
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Place a large skillet over medium heat. Add the oil and onions. Cook, stirring occasionally, until the onions are translucent. Add the spinach and cook until completely wilted. Continue cooking, stirring frequently, until any liquid evaporates. Transfer the vegetables to a strainer; press to drain as much liquid as possible. Add the Pizza Seasoning and salt and pepper to taste.
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To make the dough: Mix and then knead all the dough ingredients together to make a smooth, slightly sticky dough. This will take about 7 minutes in a stand mixer using the dough hook. Cover and let rise for 1 to 1 1/2 hours. The dough should be quite puffy looking, nearly doubled in size.
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To assemble: Gently deflate the dough. On a lightly greased or floured piece of parchment, roll the dough into a 13”-tall x 11”-wide rectangle. Spread the cooked sausage in a 3”-wide strip down the center of the dough, leaving 1” at the top and bottom of the rectangle uncovered. Use the back of a serving spoon to evenly space five indentations in the sausage, top to bottom, to cradle the eggs.
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Place the peeled eggs in the indentations you made for them; sprinkle lightly with salt if desired. Cover the eggs and sausage with the spinach mixture and sprinkle the cheese over the spinach, making sure the two long edges of dough on either side remain bare.
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To shape the braid: Begin by using your bench knife or a sharp knife to cut and remove a 1” rectangle from each of the dough's four corners. This “notching” will make the ends of the finished braid neater. Note: To see this shaping method in action check out our video, How to make a filled braid.
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Next, fold the resulting dough "tab" over the edge of the filling on one end. Starting at that same end, cut 1"-wide crosswise strips down the length of the outside sections, making sure you have the same number of strips on each side.
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Beginning on the left, lift the top dough strip and gently stretch it across the filling diagonally. Repeat on the other side with the top dough strip, so that the two strips crisscross each other. Continue down the entire loaf, alternating strips to shape the braid. When there are just two strips remaining (one on each side), fold the remaining "tab" over the filling, then crisscross the final two strips.
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Transfer the bread, parchment and all, to a baking sheet. Cover with greased plastic or your favorite reusable cover and let rise for 30 minutes while you preheat the oven to 375°F.
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To bake: When the bread has risen and become puffy, uncover and brush all over with the egg wash. Bake for 35 to 38 minutes, until golden brown. Remove from the oven and cool for 10 minutes before slicing.
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Storage information: Refrigerate the braid, well wrapped, for up to five days; reheat to serve.