Preheat the oven to 350 degrees F (177 degrees Celsius). Spray or butter and flour in three 9-inch round baking pans. Line the bottoms with parchment rounds, and set aside.
In a food processor, pulse 1 ¼ cups of pistachios until a finely ground powder forms.
In a medium mixing bowl, whisk together the ground pistachios, flour, baking powder, baking soda, and salt.
In the bowl of a stand mixer, beat the butter on high until smooth and creamy, about 2 minutes. Add the sugar and beat for 2 minutes on high. Add the vanilla, almond, and egg; mix until combined.
With the mixing speed on low, alternate adding the dry ingredients and the milk and cream. Mix until a few dry streaks remain.
In a medium mixing bowl, using a hand mixer, beat 3 egg whites and cream of tartar until stiff peaks form. Using a rubber spatula, gently fold into the cake batter until incorporated.
Divide the batter evenly between the prepared pans. Place in the oven and bake for 35-40 minutes, or until the top springs back when touched.
Remove from the oven and allow to cool in the pans for 15 minutes before inverting onto a wire rack to cool completely.
Meanwhile, to make the frosting, in the bowl of a stand mixer, beat the butter on high until creamy, about 2 minutes. Add the vanilla, salt, and powdered sugar; mix until combined. Add the mascarpone and mix on low for 10 seconds, then fold by hand until a smooth, creamy frosting forms.
To assemble the cake, spread ¼ of the frosting between the first layer, another ¼ of the frosting between the second layer, and spread the remaining frosting over the top and down the sides of the cake. Chop the reserved ¼ cup of pistachios and sprinkle on the top edges. Add a pinch or two of large sea salt flakes before serving.