1. In large skillet, heat butter over medium heat. Add mushrooms, onion, pepper and carrots; saute until liquid evaporates and mushrooms are tender, about 13 minutes. Stir in flour, lemon juice, dill, salt and pepper; cook, stirring, 2 minutes, until thickened. Remove from heat. Stir in Parmesan cheese. Let cool.
2. Unfold 1 pastry sheet on floured work surface. Spread half of vegetable mixture over pastry. Starting on a side parallel with folds, roll up pastry halfway. Repeat with opposite side, enclosing filling and forming double pinwheel. Repeat with remaining filling and pastry sheet. Cover; chill 1 hour.
3. Heat oven to 400 degrees F. Slice pastry crosswise 1/4 inch thick, about 32 slices per roll. Place slices, cut side down, 1 inch apart, on ungreased baking sheets. Stir egg and water in small bowl. Brush just tops of slices with egg wash.
4. Bake in 400 degree F oven for 15 minutes or until golden. Serve warm.