Make the cake: Preheat the oven to 350 degrees F. Sift the cake flour, 3/4 cup superfine sugar and the salt onto a piece of parchment paper. Sift the flour mixture a second time onto another piece of parchment; set aside. Beat the egg whites in a large bowl with a mixer on medium-high speed until frothy, about 1 minute. Add the cream of tartar and continue beating until soft peaks form, about 4 more minutes. Gradually beat in the remaining 1 cup superfine sugar; increase the speed to high and beat until the egg whites are stiff and glossy, about 7 minutes. Beat in the vanilla, rosemary, thyme and lemon zest.
Using a rubber spatula, gently fold in the flour mixture, 1/4 cup at a time, until fully incorporated with no pockets of dry ingredients. Transfer the batter to an ungreased 10-inch angel food cake pan. Bake until golden and the cake springs back when pressed, 40 to 45 minutes. Invert the cake pan onto a small funnel or bottle neck and let cool completely, at least 1 hour.
Meanwhile, make the compote: Combine the brown sugar, rum, 1/2 cup water and the rosemary and thyme sprigs in a medium saucepan over medium-high heat; simmer, stirring, until the sugar dissolves, about 2 minutes. Add the pineapple, return to a gentle simmer and cook until the pineapple is softened and the liquid is syrupy, about 10 minutes. Stir in the vanilla. Transfer to a bowl and refrigerate, stirring occasionally, until cool, about 1 hour. Remove the herb sprigs.
Loosen the edge of the cake with a knife, tap the sides of the pan against the counter and unmold the cake onto a platter. Slice and serve with the compote.
Cake flour is a low-protein flour, which makes for a lighter-textured cake.