If you plan to use homemade caramel, make it the day before and refrigerate it overnight so it thickens and is easy to handle.

Prepare the brown butter first: add the cubed butter to a small saucepan.

Warm the pan over medium heat and melt the butter completely.

Continue cooking, stirring constantly, until the milk solids turn golden brown and smell nutty. Watch closely so it does not burn.

Pour the brown butter into a heatproof bowl and refrigerate until cooled and slightly thickened, then bring it to room temperature before frosting.

Preheat the oven to 350°F (180°C) and grease twelve 3-inch mini cake pans with butter or nonstick spray.

In a large mixing bowl, whisk the eggs, dark brown sugar, and granulated sugar until well combined.

Mix in the neutral oil and heavy cream until smooth.

Sift the flour, baking powder, cinnamon, vanilla bean powder, and salt into the bowl and stir just until blended.

Fold in the shredded carrots and toasted walnuts with a spatula.

Scoop two large scoops of batter into each mini cake pan to make 12 mini cakes.

For cupcakes, use one large scoop of batter per lined muffin cup to make 24 cupcakes.

Bake the mini cakes for 20 to 22 minutes, or the cupcakes for about 18 minutes, until a toothpick inserted in the center comes out clean.

Cool the cakes completely on a wire rack before filling and frosting.

Make the frosting once the cakes have cooled: place the brown butter and powdered sugar in the bowl of a stand mixer.

Beat the brown butter and powdered sugar on high speed for 5 to 8 minutes until pale and fluffy.

Add the cream cheese and vanilla bean paste, then mix on low for about 1 minute, just until incorporated.

Pour in the heavy whipping cream and beat on medium speed until the frosting is thick, smooth, and pipeable.

To assemble, cut a small well from the center of each cooled cake, spoon in the salted caramel, and pipe the brown butter cream cheese frosting over the top.