Preheat your oven to 350°F (180°C) and line a baking sheet with parchment paper. This prevents sticking and ensures even baking.
In a bowl, combine the goat cheese, cream cheese, honey, salt, and pepper. Use a fork or spatula to mix until smooth and creamy. Tip: For a more textured filling, don't overmix.
Lay one filo pastry sheet on a flat surface and brush lightly with olive oil. This prevents it from drying out and helps the layers crisp up nicely.
Spoon 2-3 tablespoons of the cheese and honey mixture onto the center of the sheet, spreading it evenly. Then, carefully roll it up, being careful not to let the filling spill out.
Shape the rolled sheet into a spiral (snail) shape and place it on the prepared baking sheet. Repeat with the remaining filo sheets.
Brush the tops with olive oil to give them a beautiful shine and golden-brown color after baking. Tip: If desired, sprinkle with a few sesame seeds (only if no one is allergic).
Bake for 35-40 minutes, or until the pastry is golden brown and crispy. Keep an eye on it towards the end to prevent burning, as the honey can cause the top to caramelize quickly.
Once removed from the oven, let it rest for 10 minutes. This will help the filling set and make the burek easier to cut.
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