Prepare the syrup by combining the sugar and water in a saucepan. Bring to a boil (do not over boil), add the rose water and lime juice, and set aside.

For the dough, combine the milk, oil, syrup, rose water, and egg in a food processor mixing bowl. Add the flour and mix well for 5 to 10 minutes to form a dough that does not stick to your hands. Divide the dough into 2 balls of equal size and immediately wrap in plastic. Set aside.

For the filling, finely grind the pistachios, sugar, cardamom, and rose water in a food processor. Set aside.

Brush a 17x11x1-inch rimmed baking sheet with ¼ cup oil. Place the oven rack in the center and preheat oven to 350°F (180°C).

Prepare and dust with flour a large, wide surface for rolling out the dough. Unwrap 1 ball of dough and roll out into a very thin rectangular layer with a thin wooden rolling pin (the rolled dough should be thinner than a pie crust). Roll dough from the center to the outside edge in all directions, giving it a quarter turn occasionally for an even thickness—dust the dough with flour as necessary. The finished dough should be larger than the baking sheet.

Roll the thin layer of dough around the wooden rolling pin and transfer it to the oiled baking sheet and unroll the dough until it covers the whole baking sheet. Do not cut off the excess dough; let the dough hang over the edge of the baking sheet, 1 inch on all sides.

Spread all the pistachio-filling mixture on top of the dough. Spread, press down, and smooth it out using your hands. Use a small rolling pin to even it out and to pack it down firmly (this is important to achieve a tight, firm baklava when baked).

Roll out the second ball of dough into a very thin rectangular layer as you did in step 5 for the first one. Transfer dough sheet on top of the filling, allowing it to hang over all sides like the bottom layer. Press down on the dough evenly with your hands and smooth out the surface of the dough.

Fold and roll the overhanging dough from the top layer under the dough from the bottom layer. Press together and pinch the top and bottom edges together to seal like a pie, forming a rim around the edge of the baking pan.

Hold down the dough with one hand, pressing down with your palm while cutting the dough with a sharp knife (all the way through) into diamond shapes. Use a brush to paint the dough with 1/4 cup oil.

Place the baking pan in the middle of the pre-heated oven and bake for 30 to 35 minutes (depending on your oven) until the baklava is golden or pinkish in color.

Spread a damp towel on the counter. When the baklava is pink, remove it from the oven and place it on the towel. Evenly pour 2 cups of the syrup all over the top.

Decorate the baklava with pistachios and rose petals. Cover immediately, while still warm, with a layer of plastic wrap and a layer of aluminum foil and seal tightly. Let stand at room temperature for at least 4 hours. It is important that the baklava be covered at all times because it dries out very easily.

When ready to serve, use a sharp knife to lift 2 of the diamond pieces out of the baking sheet (to create an opening), then carefully lift 1 diamond at a time and arrange on a serving dish. Or transfer to a covered glass container for storing in the refrigerator.