Instructions

for the cake

Adjust oven rack to middle position and heat oven to 350 degrees. Grease three 8-inch round cake pans, line with parchment paper, then grease and flour pans. Process walnuts in food processor until finely chopped, about 1 minute. Transfer ground walnuts to a small bowl and set aside.
Whisk flour, 1¼ cups sugar, baking powder, baking soda, and salt together in large bowl. Whisk buttermilk, egg yolks, melted butter, oil, and vanilla together in separate bowl. Add buttermilk mixture to flour mixture and whisk until just combined.
Using stand mixer fitted with whisk attachment, whip 3 egg whites and cream of tartar on medium-low speed until foamy, about 1 minute. Increase speed to medium-high and whip whites to soft, billowy mounds, about 1 minute. Gradually add remaining ¼ cup sugar and whip until glossy, stiff peaks form, 2 to 3 minutes.
Whisk one-third of whites into batter. Using silicone spatula, gently fold remaining whites into batter until no white streaks remain. Gently fold in ground walnuts until just combined. Divide batter evenly among prepared pans, smooth tops, and gently tap pans on counter to release air bubbles.
Bake until tops are light golden and toothpick inserted in center comes out clean, 18 to 22 minutes, rotating pans halfway through baking. Let cakes cool in pans for 15 minutes. Remove cakes from pans, discard parchment, and let cool completely on wire rack, about 2 hours.

for the candied nuts

Line rimmed baking sheet with aluminum foil and set wire rack in sheet. Simmer maple syrup in small saucepan over low heat until reduced by half, 2 to 3 minutes. Stir remaining ½ cup walnuts into syrup, then pour onto wire rack. Let sit until coating is firm, about 10 minutes.

for the frosting

Whisk sugar, maple syrup, egg whites, and maple extract together in bowl and place over medium saucepan with ½ inch of barely simmering water. Using hand mixer, whip mixture on medium-high speed until stiff peaks form, 6 to 8 minutes. Remove from heat and whip until mixture has cooled and is very thick and glossy, 8 to 10 minutes.
Place 1 cake round on serving platter. Spread 1 cup frosting over cake, then top with second cake round. Spread another 1 cup frosting over cake, then top with third cake round. Spread remaining frosting evenly over top and sides of cake. Garnish with candied walnuts. Serve.