For Green Tea sponge cake 4 tablespoons cake flour (not self-rising), sifted after measuring, plus additional for dusting pan; 2 whole large eggs at room temperature for 30 minutes; 1/4 cup green tea powder; 1/2 cup confectioners sugar, sifted after measuring; 2 large egg whites at room temperature for 30 minutes; 1/8 teaspoon cream of tartar; 1/8 teaspoon salt; 1 tablespoon granulated sugar; 2 tablespoons unsalted butter, melted, foam discarded, and butter cooled;
For green tea syrup 2 teaspoon green tea powder; 1/2 cup plus 1 tablespoon water; 1/2 cup granulated sugar; 1/4 cup Cognac or other brandy;
For green tea buttercream 2 teaspoons green tea powder; 1/4 cup plus 1 tablespoon water; 6 tablespoons granulated sugar; 2 large egg yolks
1 stick (1/2 cup) unsalted butter, cut into 1/2-inch cubes and softened;
For chocolate glaze 3/4 stick (6 tablespoons) unsalted butter; 7 oz fine-quality bittersweet chocolate (not unsweetened; preferably 70 to 71% cacao), coarsely chopped
Special equipment: a 15- by 10-inch shallow baking pan; an offset metal spatula; a candy thermometer; a small sealable plastic bag
Make sponge cake:
Put oven rack in middle position and preheat oven to 425°F. Butter baking pan, then line bottom with a sheet of parchment or wax paper, leaving a 1-inch overhang on short sides, and generously butter paper. Dust pan with cake flour, knocking out excess.
Beat whole eggs in a large bowl with a handheld electric mixer at high speed until eggs have tripled in volume and form a ribbon when beaters are lifted, 2 to 3 minutes. Reduce speed to low, then add green tea powder and confectioners sugar and mix until just combined. Resift cake flour over batter and gently fold in.
Beat egg whites in a bowl with cleaned beaters at medium speed until foamy. Add cream of tartar and salt and beat until whites just hold soft peaks. Add granulated sugar, then increase speed to high and beat until whites just hold stiff peaks.
Fold one third of whites into green tea mixture to lighten, then fold in remaining whites gently but thoroughly. Fold in butter, then pour batter evenly into baking pan, spreading gently and evenly with offset spatula and being careful not to deflate (batter will be about 1/4 inch thick).
Bake until very pale golden, 8 to 10 minutes, then cool in pan on a rack 10 minutes.
Loosen edges of cake with spatula, then transfer cake (on paper) to a cutting board. Cut cake into strips and squares. Trim outside edges slightly, then carefully peel paper from strips and squares and set back on paper.
Make green tea syrup:
Stir together green tea powder and 1 tablespoon water until powder is dissolved. Bring sugar and remaining 1/2 cup water to a boil in a 1- to 2-quart heavy saucepan, stirring until sugar is dissolved. Reduce heat and simmer syrup, without stirring, 5 minutes. Remove from heat and stir in Cognac and coffee mixture.
Make green tea buttercream:
Stir together green tea powder and 1 tablespoon water until powder is dissolved. Bring sugar and remaining 1/4 cup water to a boil in a very small heavy saucepan, stirring until sugar is dissolved. Boil, without stirring, washing down any sugar crystals on side of pan with a pastry brush dipped in cold water, until syrup registers 238°F on thermometer (soft-ball stage; see cooks’ note, below).
While syrup boils, beat yolks in a large bowl with cleaned beaters at medium speed 1 minute.
Add hot syrup to yolks in a slow stream (try to avoid beaters and side of bowl), beating, then add green tea mixture and beat until completely cool, 3 to 5 minutes. Beat in butter, 1 piece at a time, and beat until thickened and smooth.
Melt butter and all but 2 tablespoons chopped chocolate in a double boiler or in a metal bowl set over a saucepan of barely simmering water, stirring occasionally, until smooth. Remove top of double boiler and stir in remaining 2 tablespoons chocolate until smooth, then cool glaze until room temperature but still liquid.
Put 1 cake square on a plate, then brush generously with one third of green tea syrup. Spread half of buttercream evenly over top with cleaned offset spatula, spreading to edges.
Arrange both cake strips side by side on top of first layer (any seam will be hidden by next layer), then brush with half of remaining green tea syrup. Spread half of glaze evenly over top, spreading just to edges.
Top with remaining cake square and brush with remaining green tea syrup. Spread remaining buttercream evenly over top, spreading just to edges. Chill cake until buttercream is firm, about 30 minutes.
Reheat remaining glaze over barely simmering water just until shiny and spreadable (but not warm to the touch), about 1 minute. Pour all but 1 tablespoon glaze over top layer of cake and spread evenly just to edges. Scrape remaining tablespoon glaze into sealable plastic bag and twist bag so glaze is in 1 corner. Snip a tiny hole in corner and decorate cake (leave a 1/2-inch border around edges). Chill cake until glaze is set, about 30 minutes, then trim edges slightly with a sharp serrated knife.
1. Buy a good and expensive Green Tea Powder (Matcha)