Place the butter and sugar in a saucepan and place over medium heat. Stir until the butter is melted and then add the apple and cinnamon. Stir well so that all the apple pieces are covered with sugar. Simmer for 10-15 minutes until apple pieces are soft. Stir occasionally. Place apple pieces in a container and let cool. Spoon the granola through the compote.
Fill a large bowl half full of ice and add water until the ice is just covered. Put the milk, cream, sugars, vanilla, cinnamon and salt in a saucepan. Heat on medium-high heat until just about boiling. Beat the egg yolks in a heatproof bowl. Pour the hot milk into the egg yolks while stirring to temper them.
Pour everything back into the pan and put the heat low. Keep stirring with a spatula until the cream reaches 80ºC or thickens. Pour through a sieve into a bowl and put the bowl on the ice bath. Add the bourbon and keep stirring regularly until your ice cream base has cooled. Then put in the fridge for at least 4 hours, but preferably a whole night.
Pour your ice cream mixture into your ice cream maker and let it run until the ice cream is ready. Spoon the apple compote through the ice cream and place the container in the freezer at least 2 hours before serving.