For the cinnamon doughnuts, in a large bowl, mix together the flour, salt, half a teaspoon of the ground cinnamon, 40g/1½oz of the caster sugar and all of the butter, water and yeast until well combined. Keep mixing for 4-5 minutes, or until the mixture comes together as a smooth dough. Cover with cling film and set aside for one hour to prove.
When the doughnut dough has doubled in size, divide it into eight equal portions and roll each into a ball.
Line a tray with baking parchment and space out the dough balls onto it, leaving plenty of room between each ball. Set aside in a warm place to rise for a further 20 minutes.
Heat the oil in a deep fat fryer to 150C/320F. Alternatively, heat the oil in a deep heavy-based frying pan until a breadcrumb sizzles and turns brown when dropped into it. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Carefully lower the dough balls into the hot oil in batches and deep fry for 6-8 minutes, or until golden-brown. Remove from the pan using a slotted spoon and set aside to drain on kitchen paper.
When the excess oil has drained off the doughnuts, mix together the remaining tablespoon of ground cinnamon and the remaining 75g/2¾oz of sugar and sprinkle onto a plate. Dredge the doughnuts in the mixture until completely coated.
For the cinnamon ice cream, bring the milk, cinnamon sticks, ground cinnamon, vanilla pod and vanilla seeds to a simmer in a pan, stirring regularly. When the milk is simmering, remove the vanilla pod and cinnamon sticks and discard (they can be cleaned well and reused in another recipe).
Meanwhile, whisk the egg yolks and sugar together in a bowl until pale and fluffy.
Pour the milk mixture over the egg and sugar mixture, whisking continuously until well combined. Return the mixture to the saucepan and continue to cook for 2-3 minutes over a medium heat, or until thickened. Remove from the heat and leave to cool slightly.
When the ice cream mixture has cooled slightly, strain it through a fine sieve over the whipped cream. Whisk until smooth and well combined, then pour the mixture into an ice cream maker. Churn for 40-50 minutes, or until almost solid, then transfer to a freezeable container and chill in the freezer.
For the chocolate sauce, bring the sugar and water to a simmer in a pan. Simmer for 1-2 minutes, stirring continuously, until all of the sugar has dissolved.
Add the chocolate pieces and whisk until the chocolate has melted and the mixture is smooth. Remove from the heat and set aside.
To serve, place two cinnamon doughnuts into the centre of each of four serving plates. Serve a scoop of cinnamon ice cream alongside and drizzle over the chocolate sauce.