Roast the sweet potato. Preheat the oven to 425℉ and line a baking sheet with parchment paper. Scrub, peel, and dice your sweet potatoes into ½ inch cubes, then toss with olive oil and salt. Roast in the oven for 20-25 minutes, or until fork tender. Set aside to cool.
Brown the butter (for the frosting). In the meantime, add the unsalted butter to a small nonstick pan and melt over medium heat. Continue to cook, stirring frequently, until the mixture foams and begins to toast. Once the butter begins to smell nutty and turn golden-brown, immediately scrape it into a separate bowl. Set in the fridge to cool and solidify as you make the cake.
Prepare the cakes. Preheat the oven to 350℉. Grease 2 (8-inch) round cake pans with butter. Dust both tins with flour, tapping out the excess, then line the bottoms with a layer of parchment paper. Set aside.
In a medium bowl, sift together the 2 ½ cups cake flour, 1 ½ teaspoons baking soda, 1 teaspoon baking powder, 1 ½ teaspoons cinnamon, 1 ½ teaspoons ginger, ¾ teaspoon salt, and ¼ teaspoon nutmeg. Mix until evenly incorporated and set aside.
Once the sweet potato has cooled, use a fork or immersion stick to blend it into a smooth paste. Measure out 1 ½ cups of sweet potato purée and blend with the remaining wet ingredients: ⅓ cup sour cream, ⅔ cup whole milk, 2 teaspoons vanilla extract.
Separate the eggs and place the whites into a clean bowl with ¼ teaspoon of cream of tartar. Reserve the yolks for later.
Using a hand mixer or stand mixer fitted with the whisk attachment, beat the egg whites to medium peaks. Do not overmix—when you lift the beater, the meringue should hold a firm, slightly drooped peak but not look dry or lumpy. Set aside.
Using the same hand mixer (no need to wash the beaters), whip the reserved egg yolks, light brown sugar, and granulated sugar in a bowl on medium speed. With the mixer running, gradually stream in the vegetable oil. Continue to beat the mixture until ribbony, lightened in color, and emulsified, about 3 minutes.
Wit the mixer on low, add in ⅓ of the dry ingredients and mix until just incorporated. Add in ½ of the wet ingredients and mix. Continue alternating the dry and wet ingredients in three additions, ending with the dry mix.
Right before the flour is fully incorporated, add one large spoonful of whipped egg whites to the bowl and stir until well combined. Using a rubber spatula, gently fold the remaining egg whites into the batter until fully incorporated and no white streaks remain, being careful not to overmix.
Divide the batter between the prepared pans, using a silicone or offset spatula to smooth the top. Bake for 30 to 45 minutes, until a skewer inserted in the center of the cake comes out clean or with a few moist crumbs.
Let the cakes cool for 10 minutes before inverting onto a wire rack. Turn the cakes to sit right side up on the rack, leaving the parchment beneath so the cakes won't stick. Allow to cool completely before assembling.
Make the frosting. Once the brown butter has solidified, remove it from the fridge to come back to room temperature. Once malleable, place the butter into a bowl and beat with a hand mixer until smooth. Beat in the mascarpone cheese.
Add the vanilla, salt, and powdered sugar, then mix until a thick frosting forms. If it's too stiff, stir in milk or heavy cream to the desired consistency. If too thin, add more powdered sugar.
Assemble. If needed, use a serrated knife to level off the tops of the sweet potato cakes. Peel the parchment from the bottom of one cake layer and place onto a serving plate. Top the cake with a large dollop of frosting and smooth with a butter knife or offset spatula. Top with the second cake layer, then spread the remaining frosting over the top and sides. Decorate the cake as desired, then serve.