To make the syrup, put all the ingredients in a saucepan with 185 ml (¾ cup) water, bring to the boil and boil for 5 minutes or so, until it is clear and slightly thickened. Remove from the heat and leave to cool completely.
Preheat the oven to 180°C (350°F/Gas 4). Lay a sheet of filo on the work surface and brush generously with melted butter. Top with a second sheet of filo and brush with butter. Repeat with a third sheet of filo, brushing the top with butter. Cut the filo into four equal strips along its length. Do the same with the remaining sheets of filo, so you have eight strips in total.
Finely chop the apricots in a food processor. Mix the apricots and nuts in a bowl and taste to see if it is sweet enough — mix some sugar through if the apricots taste a bit sour. You should have a moist rough paste.
Whip the egg white in a bowl until soft peaks form and then whisk in the sugar and vanilla. Continue whisking until stiff and creamy.
Take a heaped tablespoon of the apricot mixture and press onto one strip of the filo about 7 cm (3 inches) up from the bottom edge. Put a heaped teaspoon of the egg white over the apricot mix, and then fold the bottom of the pastry up over the filling. Turn in both the sides, pressing these down all the way up the length. Then roll up to make a neat package and seal the filling in. Brush with melted butter and put on a tray lined with baking paper. Repeat with the rest of the pastry and filling. Bake for 15–20 minutes until the pastry is crisp and golden.
Arrange the baklava on a serving plate. Pour the cool syrup over the hot baklava and serve warm with a scoop of not-too-sweet vanilla ice cream, spooning any extra syrup over the baklava. These can also be eaten cold.