Blend the cottage cheese until completely smooth and refrigerate it. Whisk the flour, baking powder, baking soda, salt, rosemary and 1/3 cup parmesan in a large bowl.
Grate the frozen butter into the flour on the large holes of a box grater and toss to coat. Pour in the blended cottage cheese and stir with a fork until a shaggy dough forms, adding milk 1 tablespoon at a time only if needed.
Turn the dough onto a floured surface, pat it out, fold it in half and repeat once more. Pat it to 1 inch thick and cut out 2 1/2-inch rounds, pressing straight down, to make about 6 biscuits.
Chill the biscuits for 10 minutes while you preheat the air fryer to 330°F for 3 minutes. Line the basket with a piece of perforated parchment paper.
Arrange the biscuits in the basket about 1 inch apart, working in batches if needed. Air fry for 10-13 minutes, until puffed, golden and cooked through in the center.
Brush the hot biscuits with melted butter and sprinkle with the remaining parmesan so it melts slightly. Serve warm.