Blend the cottage cheese in a food processor until completely smooth, then transfer it to a bowl. Add the flour, sugar and salt to the processor and pulse to combine, then add the cold butter cubes and pulse until the pieces are the size of small peas.Add the blended cottage cheese and pulse just until the dough comes together in large clumps.
Turn the dough out, divide it into 3 equal portions and shape each into a flat disk. Wrap and chill for at least 1 1/2 hours, until firm.
In a small bowl, mix the walnuts, brown sugar and cinnamon for the filling. Preheat the oven to 350°F and line 2 baking sheets with parchment paper.
Roll one disk on a lightly floured surface into a 10-inch circle about 1/8 inch thick. Brush with 1 tablespoon of the melted butter and sprinkle evenly with one third of the filling, pressing it in gently.
Cut the circle into 12 wedges with a pizza cutter. Roll each wedge up tightly from the wide end to the point and place point side down on the baking sheet.Repeat with the remaining dough and filling, then chill the shaped rugelach for 15 minutes so they hold their shape.
Brush the tops with the beaten egg. Mix the sugar and cinnamon for the topping and sprinkle it over each rugelach.
Bake for 22 to 25 minutes, until deep golden and crisp on the outside. Cool on the baking sheets for 10 minutes, then move to a wire rack.