Preheat a skillet over medium heat. Add the olive oil.
Add the sliced onion to the heated pan and cook, stirring occasionally, for 4-5 minutes. Add the carrots and cook for 2-3 minutes. Add the dill and stir through. Remove the skillet from the heat.
Transfer the carrot mixture to a bowl and let it cool for a few minutes. Add the feta cheese. Stir. Season with salt and pepper.
Place one sheet of phyllo on the counter in front of you with the long edge towards you. Cover the rest of the phyllo with plastic wrap and a towel to help prevent it from drying out.
Brush the sheet lightly with olive oil.
Quarter the carrot mixture and place one quarter along the bottom side of the phyllo sheet. Spread it out into a line, about an inch away from the bottom edge, and an inch away from the sides.
Fold the bottom edge of the phyllo up and over the carrot mixture, and then continue rolling the phyllo into a log shape.
Turn one end of the phyllo log into itself, twirling slowly while you go, until you have a spiral shape. Place it on the prepared baking sheet.
Repeat with the rest of the phyllo sheets and carrot mixture.
Whisk the egg with a splash of water. Brush the tops of each swirl, and then sprinkle with sesame seeds.
Bake for 25 minutes, until golden brown and crispy. The filling is very hot, you can let them cool a few minutes before serving.
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Notes
Keep your phyllo covered: Phyllo dries out fast. Keep the sheets you’re not using right away covered with plastic wrap and a damp towel. This keeps them pliable and prevents cracking when you roll.Don’t overfill the rolls: It’s tempting, but too much filling makes them hard to roll and can tear the phyllo. Use about a quarter of the mix per sheet.Brush lightly with oil: You want a thin, even layer of oil on each sheet.Bake until golden and crisp: All ovens are a bit different. Start checking at 22 minutes. The tops should be evenly browned, and the bottoms should feel crisp, not soft.Let the rolls cool completely, then store in an airtight container in the fridge for up to 3 days. Reheat in a 350°F oven or toaster oven for 10–12 minutes to restore the crispy texture. Microwaving is not recommended—it makes the phyllo sogg