In a small saucepan, combine the butter, brown sugar, and cinnamon over medium heat. Cook until butter is melted and mixture begins to bubble. Remove from the heat and allow to cool while you work on the ice cream base.

Pour the sweetened condensed milk into a large mixing bowl; set aside.

In the chilled bowl of a stand mixer fitted with the whisk attachment, beat the heavy cream on high until stiff peaks form, about 3 minutes. Stir in the vanilla. With a rubber spatula, gently fold whipped cream into the sweetened condensed milk. Fold in the chopped Oreos and cinnamon chips.

Pour half of the ice cream into a freezer-safe container. Plop tablespoonfuls of half of the cinnamon roll filling evenly over the ice cream. Pour the rest of the ice cream on top, then drop spoonfuls of the remaining filling on top. Cover, then freeze for at least 6 hours, preferably over night.