While the dough is chilling, in a small bowl, mix the sugar and the ginger, and set aside. In a small saucepan, heat the clarified butter on medium-high until melted. Add the chopped rosemary and simmer for 1-2 minutes. Remove from heat and stir in the apricot jam. Add the lemon juice, and (if desired), add a little more, to taste.
Divide dough into four pieces and roll first quarter out on a floured counter into a rectangle about 12 inches wide and 8 inches long, with the wider side to you. Thinly spread the apricot filling ((about 2 to 3 tablespoons) to all but the furthest 1/4 inch from you, which seals better once rolled if bare. Sprinkle with roughly a quarter of the sugar-ginger mixture.
Roll dough from the 12-inch side in front of you into as tight as a log as you can, using your fingers to lightly seal the ends onto the log. Repeat with remaining logs.
Place log of filled dough in freezer for 10 to 15 minutes, as the rugelach will cut more cleanly once semi-firm. Trim ends from log so they have a clean shape, and slice the log into 10 to 12 even slices. Arrange on prepared baking sheets a couple inches apart from each other.
Brush tops lightly with egg wash and sprinkle with sesame seeds. Bake for 20 to 25 minutes, until golden brown on top. Individual cookies need to cool only a few minutes on baking sheet before they can be transferred to a cooling rack, and the sooner you get them off the parchment, the easier it is to separate them from the leaked sugar filling that will have pooled on the baking sheet.
Best the day they're made, but will keep for 2-3 days in a sealed container at room temperature.