Bring a pot of water to the boil over medium heat and cook the courgettes until tender, about 5 minutes. Drain in a colander set in the sink. Set aside to cool.
Meanwhile, preheat oven to 180 C / Gas 4. Grease a 20x30cm baking dish.
Combine 125g flour, 100g sugar and 115g butter in a bowl until crumbly and well combined. Press the mixture into the bottom of the prepared baking dish.
Measure out 525g cooked courgettes and place into a liquidiser. Pour in the evaporated milk and eggs. Add 300g sugar, 4 tablespoons flour, 15g butter and the vanilla extract. Pulse the liquidiser several times to combine the ingredients, then blend until the mixture is pureed, about 1 minute. Pour the mixture over the base. Combine 1 tablespoon of sugar with the cinnamon in a small bowl and sprinkle over the filling.
Bake in the preheated oven until the centre is set, about 45 minutes. Let cool and chill in refrigerator before slicing.