• Preheat oven to 350 degrees Fahrenheit
• In a large pot, over medium-high heat,  combine melted butter and sugar.  Mix until combined and then slowly pour in milk.  Add semolina and vanilla.  Cook for about 5 minutes, stirring occasionally.
• In a bowl beat together 6 eggs using a fork.  Once thoroughly beaten slowly add them to the pot.  At this point you must continuously stir the contents of the pot. Continue to cook, stirring constantly, over medium-high heat for about 10 minutes. The custard filling is done when it has the consistency of a pudding.  You can get a sense of what it looks like here. 
• If you are making this dessert with someone else, while you are stirring the custard, they could be preparing the phyllo dough.  If you are doing this alone, simply remove the custard off of the heat and proceed to prepare your phyllo.
• Butter the bottom of a 9 1/2 x 13 inch rectangular baking pan.  Place 2 sheets of phyllo on the bottom of pan.  Your sheets of phyllo will be too large to line the bottom of the pan perfectly.  This is good.  Leave one end of the phyllo sheets hanging over the long end of your pan.  Using a pastry brush, brush on some melted butter.  When brushing the butter on the phyllo sheet do so lightly.  The goal is not to saturate the phyllo with butter.  Then, add two more phyllo sheets, this time letting the phyllo overlap on the opposite end of your pan. Brush on more melted butter.  Repeat this entire process 3 more times so that you end up with eight times 2-layer phyllo sections. (So, to summarize, your bottom layer of galaktoboureko will be 16 sheets, with butter being applied between every 2 sheets, and with phyllo hanging over the two long sides of your pan).
• Pour the custard mixture into the phyllo-lined pan.  Spread it out evenly by using the back of a spoon. Now, take those overlapping sheets of phyllo, two at a time, and cover the custard with them, alternating sides. Remember to butter each set of two phyllo sheets before covering it with another layer of phyllo, as you can see here.  When you have used up all of the overlapping phyllo sheets, take the phyllo that is still in the packaging, one sheet at a time,  and fold it in half (or fold it in such a way that it fits the baking pan perfectly).  Then, add this to the top of the galaktoboureko.  Repeat this with at least 4 more sheets of phyllo, buttering between each sheet.
• Using a sharp knife score the top phyllo layers of the galaktoboureko 3 times lengthwise.  Then, working along the 1st and 3rd lengthwise scores, create a bit of a herringbone pattern.  This is not an exact science and in fact,  the only thing you really need to know is that the vents you will create by scoring the phyllo will help it to bake properly and will be your guides when it comes to cutting out serving pieces.
• Place the pan in the middle rack of the oven and bake the galaktoboureko for approximately 45 minutes.  You may need to rotate the pan mid-way through the cooking process depending upon how heat is distributed in your oven.  You will know that your galaktoboureko is done when the phyllo is a lovely golden brown and slightly puffed up.  Your custard filling will appear very loose when you remove it from the oven.  This is normal and it will set as it cools.

For the syrup

• While your galaktoboureko is baking you can prepare the syrup.  In a medium saucepan combine the water, sugar, cinnamon stick and lemon slices.  Bring to a boil and once the sugar has dissolved reduce the heat and simmer for approximately 5 minutes.  Remove from heat and allow to cool.
• When your galaktoboureko is done remove it from the oven and slowly spoon the prepared and cooled syrup onto it, while in the pan, letting each ladle of syrup to be absorbed.  You can let your galaktoboureko cool slightly before doing so, but if you pour the syrup over it while it is still hot, you get this cool (but slightly messy) effect.
• You may find that you may not use all of the syrup. Use only as much as can be absorbed by the galaktoboureko.
• Allow your galaktoboureko to cool fully at room temperature before serving. Serving it while too warm, before the custard has set, will be messy (but still delicious).
• To store it, allow it to cool fully or overnight, before placing it in the fridge, where it will keep fresh (with the help of the excess syrup) for 4-5 days.
• Enjoy!