Preheat the oven to 425°F and line a baking sheet with parchment paper. Blend all of the cottage cheese (both the biscuit and gravy amounts) until completely smooth, then transfer 1/2 cup to a small bowl for the gravy and refrigerate both.
Whisk the flour, baking powder, baking soda, salt and sugar 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 1 1/4 cups blended cottage cheese and stir with a fork until shaggy, adding milk 1 tablespoon at a time only if needed. Turn it onto a floured surface, pat it into a 1-inch-thick rectangle and fold it into thirds, repeating the folding 2 more times.
Pat the dough to 1 inch thick and cut out 2 1/2-inch rounds, pressing straight down without twisting. Place them on the baking sheet so they just touch and bake for 14-17 minutes, until tall and golden.
While the biscuits bake, cook the sausage in a large skillet over medium-high heat for 7-8 minutes, breaking it into small crumbles, until browned. Do not drain the fat, since it forms the base of the gravy; if the sausage is very lean, add 1-2 tablespoons of butter.
Sprinkle the flour over the sausage and stir for 2 minutes to cook out the raw flour taste. Slowly pour in the milk while stirring, bring it to a gentle simmer and cook for 5-7 minutes, until thick and creamy.
Reduce the heat to low and stir in the reserved blended cottage cheese, black pepper, salt and cayenne. Heat for 1 minute without letting it boil; the cottage cheese makes the gravy extra velvety and rich.
Split the warm biscuits in half and spoon the sausage gravy generously over the top. If the gravy thickens while standing, loosen it with a splash of milk.