Place a large glass or metal pan inside another larger bowl filled with ice. Add the whipping cream and vanilla bean paste to the bowl set inside the ice bath. Set aside.
Pour the milk into a heavy bottom, medium saucepan. Add 1/2 cup of sugar, salt and cinnamon. Warm over medium heat, stirring to dissolve the sugar, until the mixture almost comes to a boil.
While the milk is warming, combine the remaining 1/4 cup of sugar with the egg yolks in a medium mixing bowl. Whisk vigorously until the mixture is pale and slightly thickened. Set aside.
Temper the egg mixture by slowly drizzling a little of the hot milk mixture into the eggs while whisking constantly. Continue adding the hot milk to the egg yolks, a little at a time, until combined. Once most of the hot milk has been added to the eggs, pour the egg/milk mixture back into the saucepan. Heat on medium, stirring constantly, until the mixture coats the back of a wooden spoon or reaches 175 degrees on an instant-read thermometer.
Remove from the heat and add 2/3 cup of the boiled cider. Stir until combined. Pour the cider custard into the cream set over the ice bath. Stir occasionally (about every 5 minutes) while cooling, about 30 minutes. Cover and refrigerate the custard for at least 8 hours or overnight before processing.
Process the ice cream according to the manufacturer directions for your ice cream maker. Once processed, scoop the ice cream into a freezer safe container, layering the ice cream with the remaining apple cider syrup. Alternately, freeze the ice cream without layering with the syrup, but serve with a drizzle of syrup to taste. Freeze the container for several hours or overnight before serving.