Preheat oven to 350º.
To prepare cake, coat 2 (9-inch) round cake pans with cooking spray; line bottoms of pans with wax paper. Coat wax paper with cooking spray; set aside.
Pour 2/3 cup boiling water over 6 tablespoons tea leaves in a bowl; steep 5 minutes. Strain through a fine sieve into a bowl; cool to room temperature.
Lightly spoon flour into dry measuring cups; level with a knife. Sift together flour, baking soda, and salt; set aside.
Combine 2/3 cup cocoa and 2/3 cup boiling water, stirring with a whisk. Cool in freezer 10 minutes, and stir in brewed tea, yogurt, and vanilla.
Place granulated sugar, brown sugar, and butter in a large bowl, and beat with a mixer at medium speed until well blended (about 5 minutes). Add egg whites and egg, 1 at a time, beating well after each addition. Add flour mixture and brewed tea mixture alternately to sugar mixture, beginning and ending with flour mixture. Pour batter into prepared cake pans; sharply tap pans once on counter to remove air bubbles. Bake at 350° for 30 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes on a wire rack; remove from pans. Carefully remove and discard wax paper. Cool completely on wire rack.
To prepare icing, pour 1/2 cup boiling water over 5 tablespoons tea leaves in a bowl; steep 5 minutes. Strain through a fine sieve into a bowl; cool to room temperature. Place cream cheese in a large bowl, and beat with a mixer at medium speed until fluffy (about 1 minute). Sift together powdered sugar and 1/2 cup cocoa. Gradually add cocoa mixture and 2 1/2 to 3 tablespoons brewed tea to cream cheese. Beat just until smooth. (Do not overbeat or icing will be too thin.) Discard any remaining tea.
Place 1 cake layer on a plate; spread with 1/2 cup icing. Top with another cake layer. Spread remaining icing over top and sides of cake. Sprinkle with hazelnuts. Store cake loosely covered in refrigerator.