Preheat the oven to 375°F and line a 12-cup muffin tin with paper liners.
Make the filling: blend the cottage cheese, cream cheese, powdered sugar, egg yolk, cornstarch and vanilla until completely smooth and glossy, about 1 minute. Transfer to a small bowl and refrigerate while you make the batter so it firms up slightly.
In a large bowl, whisk together the flour, pumpkin pie spice, baking soda, baking powder and salt.
Rinse the blender, then blend the pumpkin puree, cottage cheese, eggs, brown sugar, granulated sugar, oil and vanilla for 30-45 seconds. The cottage cheese should disappear completely with no visible curds.
Pour the wet mixture into the dry ingredients and fold with a spatula just until no dry streaks remain. The batter will be thick, so do not overmix.
Divide the batter among the cups, filling each about 2/3 full. Spoon the cheesecake filling evenly over the tops, about 1 1/2 tablespoons each.
Drag a toothpick through each muffin in a figure-eight to create a swirl, then sprinkle with turbinado sugar. Bake for 20-24 minutes, until the muffins are domed and the cheesecake is set at the edges with a slight wobble in the center.
Cool in the pan for 10 minutes, then move to a wire rack to cool completely. Store leftovers in the refrigerator, where the cheesecake swirl becomes even creamier.