Step 1Heat oven to 375 degrees and place a rack in the middle of the oven. To a roasting pan, add the apple chunks, brown sugar, cinnamon stick and water, tossing to combine. Cover with aluminum foil and bake for 30 minutes. Remove the foil, toss the apples with a rubber spatula and continue to bake until the apples are fork tender and lightly caramelized on top, 15 to 20 minutes.
Step 2Transfer the apples and any residual liquid to a food processor, discarding the cinnamon stick. Puree the warm apples, then add the ground cinnamon and pulse just to combine. You should have 2 to 3 cups of roasted apple puree. Set aside to cool completely.
Step 1To make the ice cream base, in a nonreactive pot, combine the milk, heavy cream and sugar and bring to a boil. Meanwhile, in a mixing bowl, whisk together the egg yolks.
Step 2Temper the egg yolks with the hot mixture by adding a ladleful at a time while whisking the yolks vigorously, until all the hot liquid has been added. Whisk in the sour cream and 1 cup of the roasted apple puree. Strain the mixture into a separate container using a fine mesh sieve. Chill the mixture over a bowl of ice.
Step 3Place a 9- by 4-inch loaf pan in the freezer. Run the mixture in your ice cream machine following the manufacturer's instructions. Once the ice cream is spun, transfer half to the chilled loaf pan. Working quickly, dot with additional dollops of apple puree and swirl into the ice cream with the back of a spoon. Top with the rest of the ice cream and repeat the swirling step with more apple puree. Place the finished ice cream in the freezer, and let it firm up completely, at least 4 hours.