Step 1Heat the oven to 400 degrees. Cut each apple into eight lengthwise wedges and then cut each wedge in half crosswise. Put the apples in a baking dish just big enough to hold them in a single layer. Add the cinnamon stick, sprinkle over the brown sugar and dot with the butter. Bake until the apples are golden brown and soft, about 50 minutes. From time to time, spoon the melted butter and cooking juices over the apples and give them a gentle stir. When the apples are done, set them aside to cool.
Step 2In a mixing bowl, combine the milk, half-and-half, cream, sour cream, ground cinnamon and sugar and whisk until smooth and well combined. Pour through a fine strainer into the ice cream maker, and freeze according to manufacturer's instructions.
Step 3When the ice cream is firm and almost done, add the apples to the ice cream freezer, discarding the cinnamon stick but spooning in all of the cooking juices. Continue freezing until the ice cream is firm. Spoon into a chilled container, cover with plastic wrap and aluminum foil and place in the freezer for at least 1 hour. Leave at room temperature for about 5 minutes to soften slightly before serving.