Make cake: Put oven rack in middle position and preheat oven to 350°F. Butter and flour cake pan, knocking out excess flour.
With blender motor at high speed, add half of toasted almonds through top hole and finely grind (be careful not to grind to a paste). Transfer to a bowl and grind remaining almonds in same manner, transferring to bowl. Add flour, baking powder, and salt to ground almonds and whisk until combined well.
Beat eggs in a large bowl with an electric mixer at high speed until foamy, about 15 seconds, then add superfine sugar a little at a time, beating. Continue beating until mixture is thick, pale, and forms a ribbon when beater is lifted, 7 to 8 minutes in a stand mixer or 10 to 14 minutes with a handheld. Add butter in a slow stream, beating, then add milk and almond extract and beat until just combined. Reduce speed to low, then add flour mixture, mixing until just combined.
Spread batter in pan, smoothing top, then sprinkle evenly with sliced almonds. Bake until top is golden, cake begins to pull away from side of pan, and a wooden pick inserted in center of cake comes out clean, 30 to 40 minutes. Cool cake in pan on a rack 30 minutes, then run a knife around edge to loosen and invert onto rack. Turn cake right side up on rack and cool completely, about 30 minutes more.
Make strawberries in syrup: Bring water, sugar, and lemon juice to a boil in a 2- to 3-quart saucepan, stirring until sugar is dissolved, then boil, uncovered, until reduced to about 2 cups, 8 to 10 minutes. Transfer syrup to a bowl and cool completely. Stir in strawberries and rose water and macerate at room temperature 30 minutes.
Whip cream and assemble dessert: Beat cream in a large bowl with cleaned beaters until it holds soft peaks.
Cut cake into 4 squares and reserve 2 squares for another use. Carefully cut each of remaining 2 squares diagonally into quarters for a total of 8 triangles, then halve each triangle horizontally. Sandwich triangle halves with strawberries, syrup, and whipped cream.
Cooks' notes: ·Cake (not cut) improves in flavor if made at least 1 day ahead (up to 3) and kept, tightly wrapped in plastic wrap, at room temperature. ·Strawberries can be macerated in syrup up to 2 hours.