Preheat the oven to 400°F and grease a 9-inch round cake pan. Blend all of the cottage cheese (both the dough and icing amounts) until completely smooth. Transfer 1/3 cup to a small bowl for the icing and refrigerate it, then stir the vanilla into the remaining cottage cheese.
Whisk the flour, sugar, baking powder, baking soda and salt in a large bowl. Grate the frozen butter into the flour on the large holes of a box grater and toss to coat every shred.
Add the vanilla cottage cheese and stir with a fork until a shaggy dough forms, adding milk 1 tablespoon at a time only if needed. Turn it onto a floured surface and knead gently 4-5 times, just until it holds together.
Roll the dough into a 10x14-inch rectangle. Spread the soft butter all over the surface, then mix the brown sugar and cinnamon and sprinkle it evenly on top, pressing lightly so it sticks.
Starting from a long side, roll the dough up snugly into a log and pinch the seam closed. Slice it into 10 even rolls with a bench scraper or unflavored dental floss and arrange them cut side up in the pan.
Bake for 20-24 minutes, until golden brown and the center roll is cooked through. Let them cool in the pan for 10 minutes.
Whisk the reserved blended cottage cheese with the powdered sugar, melted butter and vanilla until thick, smooth and glossy. Spread the icing over the warm rolls so it melts into every swirl, and serve.