To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Combine all the ingredients in a large bowl and mix until everything comes together. Knead for 6 to 8 minutes, until the dough is smooth and somewhat elastic. It may be a bit on the sticky side; that's OK., Cover and let rise until doubled, about 1 hour. While the dough is rising, make the filling., To make the filling: Place a large skillet over medium-high heat and add the oil. When it ripples, add the mushrooms and cook, stirring occasionally, for 5 minutes, until they give up their liquid and it cooks away. Season with salt and pepper, then add the onion and continue cooking until translucent. Stir in the spinach, cook until just wilted, then remove from the heat and transfer to a large bowl., Add the herbs and diced peppers. Stir to combine; let cook to room temperature, stirring every few minutes. Once cooled, stir in the grated cheese., Preheat the oven to 375°F. Line a baking sheet with parchment., After the dough's first rise, turn it out of its bowl onto a lightly floured surface and divide it in half. Pat each half into a 12" x 8" rectangle. Working with one half at a time, spoon half of the cooled filling onto the center third of the dough, smoothing it into a strip 2" wide and 10" from top to bottom (there should be an inch-wide strip of uncovered dough at the top and bottom of the filling)., Fold the top and bottom of the dough over the filling to cover the ends. Gently fold one side of the dough up and over the filling, then roll it over the remaining uncovered dough, with the seam ending up on the bottom. Repeat with the rest of the dough and filling., Transfer each log to the prepared baking sheet. Cut 1" slits in the top to reveal the filling and allow steam to escape while baking. Cover with greased plastic and let rise for 30 minutes., After this rise, uncover and brush the stromboli with the beaten egg mixture and bake for 30 to 35 minutes, until the dough is set and just begins to take on some color at the edges. Remove from the oven and cool for 10 minutes before serving warm., Store any leftover stromboli, well wrapped, in the refrigerator for up to five days.