In a medium, heatproof saucepan combine cider, bourbon, cinnamon sticks, and cloves. Bring the mixture to a boil and then reduce the heat and simmer for approximately 45 minutes or until the mixture has reduced by half.
Remove from heat and whisk in brown sugar. Pour the bourbon apple cider syrup into a bowl and chill until cold.
In a medium, heatproof bowl, whisk egg yolks, sugar, flour, and 2 tablespoons of the milk until the mixture turns pale yellow – 2-3 minutes.
In a medium saucepan, bring remaining milk and heavy cream to a simmer.
Slowly whisk milk and cream mixture into the yolk mixture to temper it. Pour the mixture back into the same saucepan and place over medium-high heat.
Heat the mixture, whisking constantly (never allowing the mixture to boil), until a thermometer reaches 185 degrees Fahrenheit or the custard coats the back of a spoon.
Remove from the heat and pour the custard through a sieve set over a large heat-proof bowl which is placed in a bowl of ice water to stop the cooking process.
Pour the bourbon apple cider syrup through the sieve as well to strain it. Discard the cinnamon stick and cloves. Whisk the syrup into the custard until well blended.
Cover the bowl with plastic wrap and chill for several hours or overnight to allow the flavors to blend.
Process the mixture in an ice cream machine.
Store the ice cream in an airtight container in the freezer.