Preheat the oven to 400 degrees F. Line a baking sheet with parchment.
Heat 1 tablespoon oil in a large nonstick skillet over high heat. Sprinkle the beef with salt and pepper and then add the beef to the skillet and cook, turning as needed, until browned all over and cooked to medium-rare, 6 to 8 minutes. Transfer the beef to paper towels to drain, wipe the skillet clean with paper towels and return the skillet to medium heat.
Heat the remaining 4 tablespoons oil in the skillet, then add the mushrooms. Sprinkle with salt and pepper and cook, stirring, until golden brown, 8 to 10 minutes. Add the onions and peppers, then sprinkle with salt and pepper and cook, stirring, until lightly charred and soft, 10 to 12 minutes. Transfer the vegetables to a bowl and let cool.
Meanwhile, place the cooled beef in the freezer for 20 minutes to firm up. Working quickly, very thinly cut the beef into the thinnest possible slices. (Slices should be almost as thin as deli meat.)
On a lightly floured work surface, roll the dough into a 20-by-14-inch rectangle. Starting on the shorter end of the dough, arrange the beef slices evenly over half of the dough, leaving a 1-inch border on the sides. Sprinkle both cheeses over the meat, then scatter the cooked vegetables evenly over the cheese. Working from that short side, tightly roll the dough up, jelly roll-style, into a log. Pinch the open seams together to seal, then tuck them underneath the log.
Transfer the log to the prepared baking sheet and brush with the egg wash. Bake until the bread is golden brown, 35 to 40 minutes.
Let the bread cool for at least 10 minutes before slicing. Halve the bread lengthwise, cut each half crosswise into 4 equal pieces and serve warm.
To make rolling and shaping the dough easy, keep the dough refrigerated until 10 to 15 minutes before you are ready to use it.
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