Preheat the oven to 220°C/gas 7. Have ready an 18cm tarte tatin dish, or ovenproof omelette pan.
Roll out the pastry on a lightly floured surface. Then, using a plate that is slightly larger than your pan as a template, cut out a circle of pastry. Lightly prick with a fork and chill while you prepare the remaining ingredients.
Lightly crush the cardamoms so that you can remove the black seeds. Discard the husks and lightly crush the seeds. Mix half of them with the sugar and set the remaining half aside. Using your fingers, press the butter on to the base of your frying pan until it coats it evenly. Sprinkle the cardamom sugar over the butter and set aside.
Place the liqueur (if using) and lemon juice in a large bowl. If you are not using any alcohol, add a little extra lemon juice to help prevent the fruit discolouring. Prepare the pears one at a time. Peel, cut into quarters and cut out the cores. Toss in the lemon juice.
Take the pears from the juice and tightly pack in a circle in the frying pan, ensuring that their more attractive rounded sides are pressed lightly into the sugar. Place on a medium-high heat. The pears will shrink slightly as they cook - do not be afraid to add another slice or two to the tarte.
Watch the pan carefully at this stage, moving it around if one area is browning faster than another. As soon as the sugar has caramelised, scatter with the remaining cardamom seeds and remove from the heat. Quickly press the pastry circle on to the pears, tucking the edges down the side of the pan, then place in the centre of the oven.
Bake for 25 minutes or until the pastry is a beautiful golden colour and well risen. Do not worry if some of the caramelised juices bubble out. Remove from the oven and leave to sit for 5 minutes. Then take a warm serving plate, press against the pastry and invert the pan, giving a good shake. The tart should slip out, juices and all. Serve hot with lots of thick cream.