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Preheat the oven to 350°F (180°C) and center the oven rack. Grease, line with parchment and flour three round 8-inch pans. (I use Parchment Paper Circles for ease.) Put two of the pans on a baking sheet (you will bake two layers then the third layer afterwards).
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Sift the flour, baking powder and salt into a medium-sized bowl. Whisk together the milk, eggs, cherry juice/syrup, almond extract and cherry flavor oil (if using) in a medium bowl or large glass measuring cup.
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In an electric mixer fitted with the whisk or paddle attachment (I used the whisk), cream the butter and sugar on medium speed until very pale and creamy, about 5 minutes.
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With the mixer still on medium speed, alternate additions of the flour mixture and milk-egg mixture, beginning and ending with the flour mixture (3 dry additions, 2 wet), beating after each addition until incorporated. Continue mixing on medium speed for 2 minutes. Fold in finely chopped cherries.
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Divide batter evenly among 3 cake pans, smoothing the surface with a small offset spatula or rubber spatula. Use a digital kitchen scale for accuracy (mine were 560 grams per pan + one 60 gram cupcake tester), if possible. Bake until a skewer comes out with a few crumbs only, about 30 minutes. Cakes should be well-risen and springy to the touch.
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Transfer the cakes to cooling racks and cool for 5 minutes, then loosen the edges by running a knife around the sides. Gently turn out the cakes, peel of parchment paper bottom, then cool right side up. Bake the third cake and repeat.