Bring 1 3/4 cups sugar and water to a boil in a 3-quart heavy saucepan over medium heat, stirring until sugar has dissolved, then wash down any sugar crystals from side of pan with a pastry brush dipped in cold water. Boil, without stirring, until it registers 220 to 225°F, 15 to 20 minutes.
At this point, while continuing to boil syrup, beat whites with espresso powder, vanilla, cream of tartar, and salt in mixer at medium speed until they just hold soft peaks. Add remaining 1/4 cup sugar, 1 tablespoon at a time, beating, and beat until whites just hold stiff peaks.
When syrup reaches soft-ball stage (238 to 242°F), immediately pour syrup in a slow stream down side of bowl into whites (avoid beaters) while beating at high speed. Beat until completely cool, 25 to 30 minutes. With mixer at medium speed, add butter 1 tablespoon at a time, beating well after each addition (see cooks’ note, below) and until buttercream is smooth. (Mixture may look curdled before all butter is added but will come together at end.)
Transfer 2 cups buttercream to a small bowl and stir in chocolate. If buttercreams are too soft to spread, chill, stirring occasionally.
Make cake layers:
Preheat oven to 350°F with rack in middle. Butter 1 sheet pan and line bottom with parchment paper, then butter parchment. Dust with flour, knocking out excess.
Whisk together yolks, milk, vanilla, and 1/2 cup sugar in a large bowl until combined well, then whisk in flour and salt until smooth. (Batter will be thick.)
Beat whites with an electric mixer until they just hold soft peaks. Beat in remaining 1/4 cup sugar, 1 tablespoon at a time, and beat until whites just hold stiff peaks.
Fold one third of whites into batter to lighten, then fold in remainder gently but thoroughly.
Spread batter evenly in pan and rap against counter to release any air bubbles. Bake until cake is dry to the touch and pale golden, 10 to 11 minutes. (Leave oven on.) Cool completely in pan on a rack.
Halve cake crosswise, cutting through parchment, to form 2 (10-by 7 1/2-inch) layers.
Prepare soufflé layers while cake bakes:.
Line second sheet pan with parchment paper.
Melt chocolate with water, then cool to lukewarm.
Beat yolks, salt, and 1/4 cup sugar in a large bowl with an electric mixer at high speed until thick and pale, about 5 minutes with a stand mixer or 8 minutes with a handheld. Fold in melted chocolate.
Beat whites with cleaned beaters until they hold soft peaks. Beat in remaining 1/4 cup sugar, 1 tablespoon at a time, and beat until whites just hold stiff peaks, about 5 minutes.
Fold one third of whites into chocolate mixture to lighten, then fold in remainder gently but thoroughly. Spread batter evenly in lined sheet pan.
Bake soufflé layers:
Bake until puffed and a wooden pick inserted into center comes out with a few crumbs adhering, 25 to 30 minutes. Transfer pan to a rack, then cover top of soufflé with 2 layers of damp paper towels. Let stand 5 minutes. Remove towels and cool soufflé completely in pan (soufflé will deflate as it cools). Sift cocoa over soufflé, then loosen edges with a sharp knife.
Halve soufflécrosswise, cutting through parchment, to form 2 (10-by 7 1/2-inch) layers.
Make meringue layers:
Toast hazelnuts, then cool, wrapped in a kitchen towel, and rub off any loose skins.
Reduce oven to 250°F
Finely chop nuts.
Beat whites with salt and cream of tartar using electric mixer until they just hold soft peaks. Beat in sugar, 1 tablespoon at a time, and beat until meringue is stiff but still glossy.
Line bottom of third sheet pan with parchment. Put small dabs of meringue under corners of parchment to secure to baking sheets.
Fold nuts into meringue and spread evenly in pan. Bake until set and pale golden, 25 to 30 minutes.
Halve meringue crosswise, cutting through parchment, to form 2 (10-by 7 1/2-inch) layers. Return to oven and bake until crisp, 45 minutes to 1 hour more. Cool in pan, then peel off parchment.
Make syrup and assemble cake:
Bring water, sugar, and espresso powder to a boil, stirring until sugar has dissolved. Cool.
Loosen edges of 1 cake layer with a knife and invert onto a flat platter. Carefully peel parchment from cake and brush with some of syrup. Spread with 1 1/4 cups mocha buttercream.
Top with 1 meringue layer and spread with 1 1/4 cups coffee buttercream.
Carefully invert 1 soufflé layer onto buttercream and peel off parchment, then gently spread with 1 1/4 cups coffee buttercream.
Repeat layering, ending with coffee buttercream (there will be some left over). Chill at least 1 hour (after that, wrap in plastic wrap). Trim all around cake with a long sharp knife to neaten edges. Bring to room temperature (about 1 hour) before serving.