Position an oven rack in the lower third of the oven and preheat to 325°F (163°C). Do not grease, butter, or flour the tube pan — the batter must cling to the ungreased sides to rise.
Whisk together the cake flour and ¾ cup granulated sugar, then sift the mixture three times through a fine mesh sieve. Set aside.
Separate 12 large eggs, placing only the whites into a clean, completely grease-free mixing bowl. Let them sit at room temperature for 20–30 minutes if they came from the fridge.
Beat the egg whites on medium speed until foamy, then add the cream of tartar and salt. Continue beating until the whites reach soft peaks, about 3–4 minutes total.
With the mixer running on medium, gradually add the remaining ¾ cup sugar one tablespoon at a time, then add the vanilla and almond extracts. Increase speed to medium-high and beat until the whites reach stiff, glossy peaks.
Sift one-quarter of the flour mixture over the egg whites and fold in gently with a rubber spatula using wide, sweeping strokes. Repeat with the remaining flour in three more additions, folding just until no dry streaks remain.
Spoon the batter gently into the ungreased tube pan, then run a thin knife through it in a zigzag pattern to release air pockets. Smooth the top with a spatula.
Bake on the lower rack for 40–45 minutes, until the top is golden and a skewer inserted in the center comes out clean. Do not open the oven during the first 30 minutes.
Immediately invert the pan onto its feet or over the neck of a tall bottle and let the cake cool completely, at least 1 to 1.5 hours. Do not skip this step — the cake will collapse if cooled right-side up.
Run a thin knife around the outer edge and center tube to loosen the cake, then push up the removable bottom and transfer to a serving plate. Slice with a serrated knife and serve with fresh berries and whipped cream.