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1
Put oven rack in middle position and preheat oven to 350u00b0F. Butter and flour cake pan, knocking out excess flour.
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2
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.
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3
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.
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4
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.
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5
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.
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6
Beat cream in a large bowl with cleaned beaters until it holds soft peaks.
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7
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.
