1. Peel and thinly slice the pears. Mix the sliced pears with both sugars in a bowl. Let sit for one hour, then purée the pear mixture in a food processor or blender until smooth.
2. Preheat the oven to 350°F. Line two 8-inch round cake pans with parchment paper and grease with nonstick spray.
3. In a large bowl, whisk together the flour, baking soda, cinnamon, salt, and nutmeg. Add the pear purée, eggs, oil, and vanilla and stir just until combined.
4. Pour the batter into the prepared pans and bake for 30–33 minutes, or until a toothpick inserted in the center comes out with just a few crumbs. Cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.

Brown Sugar Buttercream

1. In a small saucepan, whisk together the flour and milk until smooth. Cook over medium heat, whisking constantly, until the mixture thickens to a pudding-like consistency, about 5–7 minutes. Transfer to a clean bowl, press plastic wrap directly onto the surface, and cool completely to room temperature. (Do not rush this step — the mixture must be fully cool before combining with the butter.)
2. In the bowl of a stand mixer, beat the butter and brown sugar together on medium-high until very smooth and fluffy, 4–5 minutes. The brown sugar granules should be fully dissolved — the mixture should feel smooth, not gritty, when rubbed between your fingers.
3. Add the cooled flour-milk mixture to the butter mixture a spoonful at a time, beating on high after each addition. The frosting will look curdled at first — keep beating and it will come together into a silky, whipped cloud. Add the vanilla extract and a pinch of salt and beat until fully incorporated.
4. Frost the cooled cake layers as desired.