Preheat the oven to 425 degrees F. Line a baking sheet with parchment and set aside.
Combine the cauliflower florets, Brussels sprouts and olive oil on a rimmed baking sheet. Sprinkle with salt and pepper and toss until evenly coated. Bake, stirring halfway through, until lightly charred and tender, about 40 minutes. Transfer the baking sheet to a rack and let the vegetables cool. Reduce the oven temperature to 400 degrees F.
Toss the Cheddar, gouda, mozzarella, paprika and cayenne in a medium bowl until evenly combined. Set aside 1/2 cup of the cheese mixture in a separate bowl.
On a lightly floured work surface, roll the dough into a 20-by-14-inch rectangle. Spread the cheese mixture evenly over half of the dough, starting from the shorter end, leaving a 1-inch border on the sides. Scatter the roasted vegetables evenly over the cheese. Working from the 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, brush the dough all over with the egg wash and sprinkle with the reserved cheese mixture. Bake until the bread is golden brown and the cheese is browned on top, about 40 minutes.
Let the bread cool for 10 minutes. 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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