Preheat the oven to 350°. Butter and flour a 9-by-2-inch round cake pan. In a small bowl, sift together the flour, baking powder and salt. Whisk to combine.
In the bowl of a standing electric mixer, whisk together the eggs, egg yolks and granulated sugar. Set the bowl over a saucepan of barely simmering water; don't let the bottom of the bowl touch the water. Whisk vigorously until the mixture is warm to the touch. Return the bowl to the mixer and beat at medium high speed for 2 minutes. Add the vanilla and continue beating until the mixture is the consistency of whipped cream, 2 to 3 minutes longer.
Meanwhile, in a small saucepan, melt the butter in the milk; do not boil. Remove from the heat; whisk.
Sift one-third of the dry ingredients over the whipped eggs and gently fold in. Repeat this procedure twice. Pour the warm milk into the batter in a slow stream and continue folding until just incorporated.
Scrape the batter into the prepared pan and bake for about 30 minutes, or until a tester inserted in the center comes out clean. Cool in the pan on a rack for 2 minutes, then turn the cake out to cool completely.
Preheat the oven to 425°. Spread the hazelnuts on a baking sheet and toast in the oven for about 8 minutes, or until fragrant. Wrap the hot nuts in a kitchen towel and vigorously rub them together to remove most of the skins. Let the nuts cool completely. Lower the oven temperature to 225°.
Line 2 large cookie sheets with parchment paper. Using the bottom of the 9-inch cake pan as a guide, trace two circles on one sheet of paper and a third on the other.
In a food processor, finely grind the hazelnuts with the confectioners' sugar and cornstarch.
Put the egg whites in the large bowl of a standing electric mixer and set it over a saucepan of barely simmering water; don't let the bottom of the bowl touch the water. Whisk the whites vigorously until warm to the touch; don't stop for even a few seconds or the whites will cook. Then beat the egg whites with the mixer at high speed until foamy. Add the cream of tartar and beat until soft peaks form. Add the superfine sugar, 1 tablespoon at a time, beating for 10 seconds between each addition. The meringue will be stiff and glossy. Using a large rubber spatula, gently fold the nut mixture into the whites.
Scoop the meringue into a pastry bag fitted with a #5 open star tip (about 1/2 inch). Pipe a ring of meringue on one of the outlined circles on the cookie sheets 1/4 inch in from the perimeter. Then, starting in the center of the circle, pipe a continuous spiral to meet the outer ring. Fill in any holes with more meringue. It doesn't have to be perfect. Repeat to fill in the other two circles. Bake the meringues for 2 hours, or until dry and firm.
Set aside 8 perfect strawberries. Slice the remaining berries 1/4 inch thick. Toss the berries in a bowl with the superfine sugar and vanilla. Set aside, stirring occasionally.
In a small chilled bowl, whip the heavy cream with the confectioners' sugar until slightly firmer than soft peaks. Refrigerate until ready to use.
Assemble the cake no more than 3 hours before serving. Split the sponge cake into 2 even layers. Invert the top layer onto a 9-inch cardboard cake round or serving platter.
Drain the berries, reserving the juice. Using a pastry brush, dab half the juice on the cake layer. Evenly distribute half the sliced strawberries on the cake and spread 1 heaping cup of the whipped cream on top. Place a dacquoise disk over the cream and spread with 1/2 cup of the whipped cream. Repeat the layering with the second cake layer, the remaining berry juice and berries, another heaping cup of whipped cream and a second dacquoise disk.
Using a serrated knife, quarter the remaining dacquoise disk, and then cut each quarter in half to get 8 wedges. If a wedge breaks, just glue it with a little whipped cream. Whisk the remaining whipped cream until it's stiff and transfer to a pastry bag fitted with the #5 star. Pipe 8 evenly spaced rosettes of whipped cream about 1/2 inch from the edge of the cake. Pipe 1 rosette in the center. Lay a reserved strawberry on each rosette and lean a wedge of the dacquoise against each berry with the point toward the center of the cake. To serve, slice with a serrated knife.
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