Preheat the oven to 350°F and grease an 8x8-inch baking dish.
Add all of the oat ingredients to a blender and blend until smooth and pourable, about 1 minute. Pour the batter into the baking dish.
Stir together the melted butter, brown sugar and cinnamon for the swirl. Drop spoonfuls over the batter, then drag a skewer or the tip of a spoon through it to create swirls.
Bake for 25 to 30 minutes, until the center is set and springs back lightly when touched.
While the oats bake, blend the cottage cheese, cream cheese, powdered sugar and vanilla until completely smooth and creamy.
Let the baked oats cool for 10 minutes, then drizzle the icing over the top. Slice into squares and serve warm.