Make the cake: Preheat the oven to 350 degrees F. Grease two 9-inch round cake pans and line the bottom of each with parchment.
Whisk together the flour, baking powder, cinnamon and salt in a medium bowl. Whisk the egg white, milk and vanilla together in another medium bowl. Beat the butter and sugar in a large bowl with an electric mixer on medium-high until light and fluffy. Add about a third of the flour mixture and beat on medium-low until incorporated. Now add a third of the egg-milk mixture and continue to beat on medium-low until incorporated, scraping down the side of the bowl as needed. Repeat with the remaining flour and egg-milk mixtures, finishing with the flour. Divide the cake batter evenly between the prepared cake pans and bake until the cake bounces back when pressed and a toothpick inserted in the center comes out clean, 25 to 30 minutes. Let cool completely in the cake pans on a rack.
Make the buttercream: Bring a few inches of water to a boil in a saucepan that can hold a heatproof stand mixer's bowl above the water.
Add the 3/4 cup sugar, cream of tartar, salt and egg whites, to the mixing bowl and whisk together by hand. Set the bowl above the boiling water and whisk until the mixture is warm to the touch and the sugar completely dissolves. Transfer to the stand mixer fitted with the whisk attachment and beat at medium-high speed until cool and the whites hold stiff peaks, 10 to 15 minutes.
Toss in a couple of chunks of butter at a time, making sure the pieces are incorporated before adding more. After all the butter is added, continue whisking on medium-high speed. The mixture will deflate and appear curdled. Continue whisking until the buttercream comes back together to form a smooth and spreadable consistency. Beat in 1 teaspoon cinnamon and vanilla.
Combine the remaining 1 tablespoon sugar and 1/2 teaspoon cinnamon together in a small bowl.
To assemble: Slice each cake in half horizontally for a total of 4 cake rounds (set aside one of the bottom pieces for the top layer). Put 1 round on a cake stand or serving platter, and spread about 1 cup of the buttercream on top, all the way to the edge. Sprinkle with some of the cinnamon-sugar. Repeat with 2 more layers of cake, buttercream and cinnamon-sugar. Place the reserved cake round on top (browned-side up). Frost with the remaining buttercream and sprinkle with the remaining cinnamon-sugar. Let sit for at least 1 hour before slicing and serving.
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