For the filling, mash the feta, egg, mint, oregano, black pepper, and nutmeg in a mixing bowl until a paste forms. Refrigerate until ready to use.
Place the phyllo dough on a work surface and cover with a damp tea towel.
Take 1 sheet of dough, brush it with oil, and cut it lengthwise into four strips with scissors.
Place 1 generous tablespoon of filling on the corner of each long strip. Fold the short side over to a long side.
Continue folding the börek from corner to corner until a triangle forms, tucking any uneven ends under.
Brush the triangle with oil and place it on a baking sheet. Repeat until all the filling and dough is used.
Bake the börek for 20 minutes in a preheated oven at 180°C (Gas Mark 4), or deep-fry them in batches for 1-2 minutes per side in olive oil and then drain on paper towels.