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For the stromboli dough:
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1.
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In a 1 cup measuring cup, combine the water and sugar stir to combine and sprinkle the yeast over it.
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Let sit until the mixture is foamy, about 10 minutes.
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If the yeast doesnt proof (get foamy) pour it out and start again it is important to have a live yeast mixture.
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2.
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Meanwhile place the 1 cup of flour and the salt into a sifter and sift it into the medium bowl of a stand mixer.
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Make a well in the center of the flour and pour in 2 tablespoons of extra-virgin olive oil and pour in the proofed yeast mixture.
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Turn the mixer, fitted with dough hooks, to high (labeled knead on my mixer) and turn on.
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Continue kneading (mixing) until the dough comes cleanly away from the sides of the bowl and forms a ball around the dough hooks (scrape the sides of the bowl to move flour into the path of the hooks as needed).
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3.
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Turn the dough out onto a lightly floured surface and knead until smooth but still slightly tacky (add flour gradually as needed to reach the slightly tacky state), 3 to 5 minutes.
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4.
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Lightly oil a large mixing bowl with about 1 1/2 teaspoons olive oil.
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Place the dough in the bowl and turn to oil the ball on all sides.
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Cover the bowl with plastic wrap or cloth and set in a warm, draft-free place until nearly doubled in size, 1 to 1 1/2 hours.
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For the stromboli construction and baking:
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1.
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Preheat the oven to 425 degrees F and lightly oil a large baking sheet and set aside.
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2.
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In a large skillet heat 1 tablespoon of extra-virgin olive and cook the sausage over medium-high heat until browned and the fat is rendered, about 5 minutes.
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Remove with a slotted spoon and drain well on paper towels.
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Discard all but 1 tablespoon of fat from the pan.
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3.
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Add the onions and cook over medium heat until onions begin to caramelize.
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Add the balsamic vinegar, bell peppers and cook, stirring, until very soft, 4 to 5 minutes.
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Add the garlic and Italian seasoning and cook, stirring, for 1 minute.
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Taste and adjust seasoning to taste.
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Remove from the heat and cool.
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4.
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Meanwhile, remove the stromboli dough from its bowl and divide into 2 equally sized portions.
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Form each portion into a ball.
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Lightly flour your work surface and, working with one ball at a time.
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Flatten the ball into a disk.
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Using your hands and a rolling pin, work the disk into a thin (about 1/8 thick) larger disk about 10 to 11 inches in diameter (as if you were making a thin, round pizza).
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Using a 10 dinner plate as form, center the plate on the dough disk and using a sharp knife trim and discard the excess dough, remove the plate.
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You should now have a perfectly round 10 disk to work with.
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5.
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Working with the 10 disk in front of you, and using a pastry brush, brush olive oil over the surface of the dough leaving a 1/2-inch margin uncoated on the edge of the disk.
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Overlapping the slices slightly, layer half of the ham, salami, pepperoni, provolone and mozzarella cheese slices, and the onions and olives over the bottom half of the dough circle respecting the clean, 1/2-inch margin along the edge.
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6.
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Spread half of the cooled sausage mixture evenly over the cheese and sliced meats again respecting the 1/2-inch clean margin.
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Lightly brush the 1/2-inch clean margin with some of the beaten egg/water mixture and fold the upper half of the disk over the lower half pressing the edge together to seal (the tines of a fork can be used to make a crimped edge for visual appeal).
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7.
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Remove the newly formed stromboli to the lightly oiled baking sheet and repeat steps 4 thru 6 with the remaining dough ball.
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8.
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Lightly brush the tops of each of the stromboli with some of the beaten eggs being careful not to allow any of the egg to seep under the stromboli (this will tend to cause the stromboli to stick to the baking sheet as they cook and make removal difficult).
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You may cut 2 or 3 1-inch slits into the top of each stromboli to allow steam to escape if you think it necessary.
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9.
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Place the baking sheet containing the stromboli in the center of the preheated oven and bake until the stromboli are nicely, golden browned, 30 to 40 minutes.
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Serve immediately if youre going to eat the stromboli with a fork and knife or allow to cool slightly if you are going to eat them as sandwiches (to avoid burning your mouth).
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The stromboli may be served accompanied by a spicy marinara sauce for dipping.
