Preheat oven to 450º F. Butter and flour the sides of an 11-by-17 inch jelly roll pan (rimmed baking sheet). Line bottom of pan with a sheet of parchment paper cut to fit bottom pan exactly.
Beat eggs, egg yolks and ¾ cup sugar together in a medium bowl until thick and light. (This is where you want to beat for a very long time, until the mixture triples in volume.) Beat in the vanilla.
In a separate bowl, beat the egg whites until foamy, ad cream of tartar and beat until whites begin to form peaks. Add the 2 tablespoon sugar and beat until the whites hold stiff, glossy peaks. (Do not over beat.)
Sift flour over the egg yolk mixture and fold in. Stir in one fourth of the whites. Then carefully fold in the remaining whites.
Spread batter evenly in pan. Bake until light brown and springy to touch (7 to 10 minutes). Cool in pan 5 minutes. Run a knife along edge to loosen. Invert cake tin to cut out 8-¼ inch circles of cake. Wrap the cake layers, separated with waxed paper, and set aside. Cake may be frozen at this point.
Combine water and the ⅓ cup sugar in saucepan; bring to a boil to dissolve sugar. Cool to room temperature; flavor with liqueur. Set aside or refrigerate in glass jar until ready to use.
Strawberry Bavarian Cream
Sprinkle the gelatin over the strawberry puree in a small bowl and set aside until spongy.
Combine egg yolks and sugar in a bowl and beat until light. Bring milk to a boil in sauce pan. Pour hot milk into yolk mixture and stir with a wooden spoon. (It doesn't say so but I would temper the egg mixture first to be safe.) Return this mixture to the saucepan and cook over medium heat, stirring constantly, until your finger leaves a clear trail in sauce when drawn across the back of the spoon. (Do not boil or mixture will curdle.) Immediately remove from heat and stir in softened strawberry gelatin mixture. Pour into a stainless steel bowl places over a bowl of ice water. Stir in lemon juice and a few drops of red food coloring. Cool over ice water, stirring occasionally, until mixture thickens to the consistency of softly whipped cream.
White gelatin mixture is cooling, whip the whipping cream until it holds soft peaks. When the gelatin mixture resembles softly whipped cream, fold the whipped cream into the gelatin mixture.
Wash and hull strawberries; coarsely chop. Place strawberries in saucepan; crush to start juices flowing. Place over low heat; add sugar and water; simmer slowly 10 minutes. Pour juice and pulp through damp jelly bag or cheesecloth-lined colander and drain into a bowl for 15 minutes. (Do not press the fruit.)
Brush sides of 10-inch springform pan lightly with flavorless salad oil or almond oil. Cut out a cardboard circle that is exactly the same size as the bottom inside of the pan; cover cardboard with aluminum foil and fit into bottom of pan. Center one layer of the cake bottom of pan. Brush the cake with some of the soaking syrup to just moisten (not drench) the cake; set aside.
Immediately pour about half of the Bavarian Cream over the first layer of cake in the pan. Set the next layer of cake on top of the cream. Pour remaining Bavarian Cream over cake and smooth top of the cream with spatula. Refrigerate until the cream sets, 1 to 2 hours.
Place lemon juice, kirsch, and water in a small bowl. Sprinkle gelatin over this mixture; set aside until spongy and soft.
Measure 1-½ cups strawberry juice into a small saucepan and bring to a simmer; pour over gelatin mixture and stir to dissolve gelatin. Tint to desired color with red food coloring. Place bowl over bowl of ice water and stir occasionally until the mixture is syrupy and just beings to thicken (do not let jell); remove from ice water.
When mixture is syrupy, pour a 1/16-inch layer (I recommend ¼-inch or more) over the top of the cake. Refrigerate until set.
To serve: Wrap a hot towel around the outside of springform pan for a few minutes. Run a small sharp knife tip around the edge of the Strawberry Mirror to separate it form the sides of pan. Mirror will tear when sides are unlatched if it is stuck at ANY point. Slowly unlatch the pan and slide it off the cake. Slice cake in wedges and serve in upright slices.