Place pastry sheet on work surface. Cut in half, forming two 9½ x 4¾-inch rectangles. Arrange half of prosciutto on 1 rectangle, leaving a ½-inch border along 1 long side. Sprinkle prosciutto with half of basil, then top with half of cheese. Brush plain border with egg glaze. Starting at long side opposite border, roll up pastry jelly-roll style, pressing gently to seal long edges. Wrap in plastic. Repeat with remaining pastry, prosciutto, basil, cheese, and egg to form second log. Refrigerate until firm, at least 3 hours and up to 2 days.

Position rack in center of oven and preheat to 400°F. Line 2 large baking sheets with parchment paper. Using a serrated knife, cut logs crosswise into ½-inch-thick rounds. Arrange rounds on prepared sheets, spacing 1 inch apart. Bake 1 sheet at a time until pastries are golden brown, about 16 minutes. Using metal spatula, transfer pastries to racks and cool slightly. Serve warm.