Make the frosting: Whisk the egg whites and maple sugar in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water) until the mixture is warm and the sugar dissolves, about 5 minutes. Remove the bowl from the pan and beat with a mixer on high speed until stiff, shiny peaks form, about 5 minutes. Beat in the butter a few pieces at a time, then continue beating until smooth. (Don't worry if the mixture looks separated at first; keep beating and it will come together.) Beat in the maple extract.
Assemble the cake: Poke holes all over the tops of the 2 cake layers using a toothpick. Spoon the reserved walnut syrup over the cakes, letting it soak into the holes.
Coarsely chop the walnuts. Transfer 1 cup frosting to a medium bowl; fold in the chopped nuts. Put 1 cake layer on a platter and spread the walnut frosting on top. Top with the second cake layer. Cover the whole cake with the remaining maple frosting. Decorate with more walnuts.
Basic Vanilla Cake
Preheat the oven to 350 degrees F. Butter two 9-inch-round cake pans and line the bottoms with parchment paper; butter the parchment and dust the pans with flour, tapping out the excess.
Whisk 3 cups flour, the baking powder and salt in a bowl until combined. Beat 2 sticks butter and the sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the mixer speed to medium; beat in the eggs, one at a time, scraping down the bowl as needed. Beat in the vanilla. (The mixture may look separated at this point.)
Mix 1/2 cup water with the cream in a liquid measuring cup or bowl. Beat the flour mixture into the butter mixture in 3 batches, alternating with the cream mixture, beginning and ending with flour, until just smooth.
Divide the batter between the prepared pans. Bake until the cakes are lightly golden on top and the centers spring back when pressed, 25 to 30 minutes. Transfer to racks and let cool 10 minutes, then run a knife around the edge of the pans and turn the cakes out onto the racks to cool completely. Remove the parchment. Trim the tops of the cakes with a long serrated knife to make them level, if desired.