Heat the olive oil in a large sauté pan (with a lid) over high heat. Add the eggplant cubes, spreading them into an even layer, season with the salt, and cook undisturbed for 5 minutes, or until the eggplant starts to brown.

Add the onions and garlic and stir to combine. Cover the pan, reduce the heat to medium, and cook until the eggplant, onions, and garlic are soft (taking care not to burn the garlic), about 10 minutes.

Remove the lid and cook to reduce and thicken the vegetable juices a bit, about 5 minutes.

Stir in the tomatoes, paprika, sugar, and pepper and cook until the mixture is thick and nicely blended, 5 to 10 minutes, depending on how juicy your tomatoes are, stirring and scraping the bottom of the pan occasionally to prevent sticking. Taste and add more salt, pepper, and paprika if needed. Remove the pan from the heat and cool the mixture completely.

When you're ready to assemble and bake the pastries, preheat the oven to 425°F. Line two baking sheets with parchment paper.

Place one sheet of the puff pastry on a lightly dusted work surface and roll it to a 10-by-14-inch rectangle, with a short side closest to you. Trim any uneven edges. Cut the sheet lengthwise in half, so you have two 5-by-14-inch rectangles. Arrange the pastry strips on one of the prepared baking sheets. Repeat with the second sheet of puff pastry and arrange on the second baking sheet. Set a bowl of cool water nearby.

Divide half the filling evenly between the 2 pastry strips on the first baking sheets, spreading it from top to bottom along half of the width of each rectangle, leaving just less than an inch of empty border around it. Dip your finger in the water and moisten the borders to help seal the pastry. Gently fold the other half of the dough over the filling on each strip, bringing the edges of the pastry together neatly, to make a 2½-by-14-inch rectangle. Using a fork, press the edges of the rectangles to seal the pastry. every few inches to create steam vents. If necessary, reposition the pastries so they are evenly spaced on the baking sheet, and repeat with the remaining pastry strips and filling.

Brush the pastries with the whisked egg and bake until well browned and crisped, 25 to 30 minutes. Transfer to a wire rack and cool for about 10 minutes.

Cut each strip into portions on the diagonal and serve slightly warm or at room temperature, with sour cream on the side, if desired.