Step 1 Line a 13-by-9-inch baking pan with foil and grease with butter.
Step 2 Using a stand mixer fitted with the paddle attachment, cream together the room temperature butter and powdered sugar until light and fluffy, 1 to 2 minutes over low speed. Beat in one-half teaspoon salt, vanilla, zest and rosemary. Slowly incorporate 1 3/4 cups flour.
Step 3Using lightly floured hands, press the dough into the prepared pan. Refrigerate for 30 minutes. Meanwhile, heat the oven to 350 degrees.
Step 4Bake the shortbread until lightly browned, 25 to 30 minutes. Set aside and cool to room temperature.
Step 5While the shortbread is cooling, cook the apricot mixture: In a small, heavy-bottomed pan, combine the apricots, water, granulated sugar, honey, rum and remaining one-eighth teaspoon salt. Bring to a simmer and cook, stirring frequently, until thickened and most of the liquid is absorbed, about 1 hour. Set aside to cool, then puree in a food processor until smooth. Spread the cooled apricot mixture over the cooled shortbread. Heat the oven once more to 350 degrees.
Step 6In the bowl of a stand mixer, combine the dark brown sugar, almonds and remaining 3 tablespoons chilled butter with the remaining one-half cup flour, and beat until it begins to clump together.
Step 7Sprinkle the crumble topping over the apricot layer, then bake until the topping is browned, 20 to 25 minutes. Cool completely in the pan before slicing and serving.