I used homegrown duck eggs, which are a little larger than the chicken eggs they sell in stores. Just pointing this out so there's no confusion. The mixture itself also turned out much more yellow in color.
Put the eggs in a bowl and beat with a mixer (or with a whisk for a longer period of time) until the egg mixture becomes thick and voluminous. Then add the sugar and continue beating everything until the sugar melts. Add the vanilla essence. I don't use a lot because the Turkish delight itself is plenty aromatic.
Put the water and sugar for the syrup in a pot. On a medium grater, grate the rind of 1/2 a lemon directly into this, for flavor. Stir and leave it aside. You're going to be boiling the syrup on the stove until the sugar melts but do this at the end once the dessert has cooled. You will be pouring hot syrup over the cooled dessert.
You need exactly 16 phyllo pastry sheets. That's more than 14 oz (400 g) - i.e. 1 whole packet and a little more from another. Separate the 16 sheets into 2 piles of 8.
Cut each Turkish delight piece into 5 widthwise to obtain 5 squares from each. If using Turkish delight pieces that are a bit too small, cut them into 4 instead.
Use a large serving spoon to pour a spoonful from the mixture onto 1 sheet and smear it everywhere. There needs to be mixture all over the entire sheet, smear carefully. Then place another sheet on top. On it, arrange Turkish delight pieces next to each other in a row along the narrower side. Grab the ends carefully and wrap it into a nice roll.
Smear the oven tray with 3 tbsp oil. Arrange the 1st roll. Wrap all the rolls in this manner. Arrange them next to each other.
Pour 2-3 tbsp oil in a bowl. Smear the rolls with this oil using a brush. Bake in a preheated 338°F (170 °C) oven until golden.
Leave the dessert to cool. Take a ruler and cut the rolls down the middle every 6″ (15 cm). Use a sharp knife to cut carefully down the middle. Then cut each of the halves into 3″ (7.5 cm) to obtain equally-sized small baklava bites. What you get is a total of 64 small baklavas.
Boil the syrup and use a ladle to distribute it equally amongst the bites. Leave covered with aluminum foil in the fridge right in the tray. It's going to be swimming in the syrup but don't worry - it'll get absorbed entirely after it sits for a bit.
The mixture isn't runny and you've got plenty of syrup. Leave the dessert in the fridge for 1 day and then eat with coffee.