Preheat oven to 350°F (175°C). Grease three 8-inch round cake pans.

In a stand mixer fitted with the paddle, beat butter, oil, sugar, vanilla, and almond extract on medium-high for 4–5 minutes until very pale and fluffy. Scrape the bowl well.

Lightly whisk eggs together or use an immersion blender to combine. Slowly stream the eggs in, mixing well after each addition.

In a separate bowl, whisk together sifted cake flour, baking powder, baking soda, and salt.

With mixer on low, add dry ingredients in three additions, alternating with buttermilk in two additions, beginning and ending with dry. Mix just until combined.

Mix on low until batter is smooth, thick, and velvety.

Divide evenly between pans (720g-725g per pan). Bake 25–30 minutes, until lightly golden and a toothpick inserted shows a few moist crumbs.

Cool 10 minutes in pans, then turn out onto racks to cool completely.

Cream butter and sugar for frangipane until smooth.

Mix in almond flour.

Add eggs one at a time, mixing well.

Stir in extracts, flour, and salt.

Transfer mixture to a saucepan and cook over low heat, stirring constantly, about 6-8 minutes, until slightly thickened and glossy (170–175°F).

Remove from heat and cool completely. Chill until thick and pipable.

Brown 1 cup butter until golden and nutty. Pour into a heatproof dish and cool completely. Chill briefly until opaque and soft, stirring occasionally.

In a stand mixer fitted with the paddle, beat softened butter and browned butter together on medium-high for 8 minutes until very pale and fluffy.

With mixer on low, add powdered sugar and salt. Add vanilla and slowly stream in heavy cream. Beat until light and fluffy.

Level cake layers if needed.

Place first layer on a cake board. Pipe a buttercream dam around the edge.

Fill with ½ of the frangipane.

Repeat with second layer.

Top with final layer, crumb coat, and chill 30 minutes.

Apply final coat.

Pipe rosettes on top using tip 1M.

Finish with toasted sliced almonds and a dusting of powdered sugar.