Bring milk, cream, milk caramel, and cinnamon stick to a simmer in a medium saucepan over medium heat, stirring often. Remove from heat. Cover, and let stand 30 minutes. Return to a simmer over medium-high heat, stirring occasionally. Remove from heat; discard cinnamon stick. Set aside.
Prepare an ice-water bath; set aside. Put yolks and sugar into the bowl of an electric mixer fitted with the whisk attachment. Beat on medium-high speed until pale and thick, about 4 minutes.
With machine running, add hot milk mixture in a slow, steady stream. Add pumpkin and vanilla, and beat until combined. Pour mixture through a fine sieve into a bowl set in the ice-water bath. Freeze in an ice cream maker according to manufacturer's instructions. Sandwich ice cream between gingerbread cookies just before serving.