Preheat oven to 190 degrees C. Grease baking dish.
In a mixing bowl, whisk 3 eggs, white sugar, 2 tablespoons of brown sugar, 1/2 teaspoon of cinnamon, 60g of butter and 3 cups of milk together and gently stir in the bread cubes and sultanas. Lightly spoon the mixture into the prepared baking dish.
Bake in the preheated oven until browned and set in the middle, 50 to 55 minutes; cover the dish with foil after 30 minutes to prevent excessive browning. Let the pudding stand for 10 minutes before serving.
For vanilla sauce, whisk 1/2 cup of brown sugar, the flour, a pinch of cinnamon, 1 egg, 2 tablespoons of melted butter, 1 1/4 cups of milk and salt together in a heavy saucepan until smooth. Heat over medium heat, whisking constantly, until thickened and the sauce coats the back of a spoon, 10 to 12 minutes. Stir in the vanilla essence. Pour sauce over warm bread pudding or serve on the side in a bowl.