Preheat oven to 190°C/375°F. Add 1 tbsp of the olive oil, the chili flakes and the fennel seeds to a large skillet and set over medium heat. Once shimmering, add the tomato paste and garlic and cook until fragrant and the tomato paste begins to darken - 2-3 minutes.
Pour over the tomatoes along with the dried oregano and ¼ tsp of salt. Stir to combine and add in the basil sprigs. Bring to a simmer and reduce the heat to medium-low. Allow to simmer, stirring occasionally, until the sauce is thick and reduced - about 20 minutes. Remove from the heat, remove the basil sprigs and allow to cool slightly.
On a clean work surface, lay down one sheet of filo pastry and brush with some of the remaining 3 tbsp of olive oil. Lay another sheet of pastry down next to it, overlapping the ends by about 5 cm (2 in). Brush with olive oil and repeat with one more sheet of filo so that the strip of pastry measures roughly 1 metre (3 feet). Continue to brush and layer the filo so you have one more even layer.
Arrange the filo strip lengthwise across your work surface. Evenly spread the sauce over the top quarter of the pastry. Evenly sprinkle the mozzarella and the feta over the sauce.
Working carefully, tightly roll the pastry around the sauce and cheese so that it resembles a long cigar. With the seam-side facing in, gently spin the log into a spiral, tucking the end under the burek. Transfer to a parchment-lined baking sheet and brush the exterior with olive oil.
Bake the burek until the pastry is crisp and deeply golden brown - about 35-40 minutes. Allow the burek to cool for at least 10 minutes before serving.