Preheat oven to 350°F. Grease and flour the bottom and sides of three 8-inch round cake pans.
Combine milk, tea, cardamom, fennel, and ginger in a small saucepan and bring to a simmer over medium-low heat. Remove from heat and steep for 5 minutes. Strain chai milk and cool.
In a medium bowl, combine the eggs, egg yolk, vanilla, and ⅓ of the cooled chai milk and whisk to blend. Set aside.
Combine flour, sugar, baking powder, cinnamon, and salt in the bowl of a mixer and beat on low until blended. Add Plugrá Butter and beat until mixture resembles coarse cornmeal. Add remaining chai milk and beat on medium-low speed until incorporated.
On medium speed, add the egg mixture in three additions, scraping the bowl after each addition. Beat until mixture is light and fluffy.
Divide batter evenly among the pans and bake for 20 minutes, or until a cake tester inserted in the middle comes out clean.
Cool 10 minutes; remove cakes from pans and cool completely on a wire rack.
Swiss Meringue Buttercream Icing Directions
Combine egg whites and sugar in a stand mixer bowl and place over a saucepan partially filled with simmering water. Whisk constantly for about 3 minutes, or until sugar has completely dissolved.
Transfer bowl to mixer stand fitted with a whisk attachment and beat on medium-high speed for 10 minutes, or until the mixture has cooled completely and formed stiff peaks.
On medium-high speed, add Plugrá Butter, a few tablespoons at a time, mixing to incorporate after each addition. Continue mixing until smooth; add vanilla and beat until just combined.
To assemble cake: spread frosting between cake layers and over top and sides of cake.