Melt the butter in a light-colored saucepan over medium heat and add the sage leaves. Cook for 5-7 minutes, swirling often, until the foaming settles, the milk solids turn deep golden brown and it smells nutty. Lift out the crisp sage leaves and set them on a paper towel.
Pour 2 tablespoons of the brown butter into a small cup and set it aside for brushing. Pour the rest, with all the toasty brown bits, into a small shallow dish and freeze for 35-45 minutes, until completely firm.
Preheat the oven to 425°F and line a baking sheet with parchment paper. Blend the cottage cheese until completely smooth and refrigerate it, then crumble the crispy sage and whisk half of it into the flour, baking powder, baking soda, salt, sugar and pepper.
Break the firm brown butter into small pieces and rub it into the flour mixture with your fingertips until the pieces are about the size of peas. Work quickly so the butter stays cold.
Pour in the 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, fold it into thirds and repeat 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 chill for 10 minutes.
Bake for 15-17 minutes, until tall and golden. Rewarm the reserved brown butter if it has set, brush it over the hot biscuits and finish with the remaining crispy sage and flaky sea salt.