Heat the oven to 400°F. Beat the egg and water in a small bowl with a fork or whisk.
Heat the oil in a 12-inch skillet over medium-high heat. Add the peppers, mushrooms, eggplant, onion, garlic powder and basil and cook until the vegetables are tender and the liquid is evaporated. Let cool to room temperature.
Unfold the pastry sheet on a lightly floured surface. Roll the pastry sheet into a 16x12-inch rectangle. With the short side facing you, spoon the vegetable mixture onto the bottom half of the pastry sheet to within 1 inch of the edge. Top with the cheese. Starting at the short side, roll up like a jelly roll. Tuck the ends under to seal. Place seam-side down onto a baking sheet. Brush with the egg mixture.
Bake for 25 minutes or until the pastry is golden brown. Remove the pastry from the baking sheet and let cool on a wire rack for 10 minutes.