Heat the oven and prepare the cake pans: Arrange a rack in the middle of the oven and heat to 350°F. Coat 2 (9-inch) round cake pans with butter or cooking spray. Sprinkle a little flour into each pan, tilt and shake to distribute evenly, then tap out the excess over the sink.

Cream the butter and sugars: Place the butter, granulated sugar, and brown sugar in the bowl of a stand mixer fitted with a paddle attachment. Beat on medium-high speed until light and fluffy, about 3 minutes.

Add the eggs: Add the eggs and beat until fully incorporated and the mixture looks creamy and very pale yellow.

Add the dry ingredients, followed by the milk and vanilla: Beat in the flour, baking powder, and salt on low speed. Add the milk and vanilla and beat on low speed for 30 seconds. Increase the speed to high and beat for 3 minutes.

Bake the cakes: Divide the batter between the prepared pans. Bake until the tops spring back slightly when pressed and the cake pulls away from the sides of the pan, 25 to 30 minutes.

Cool the cakes: Let the pans cool on wire racks for at least 15 minutes. Flip each pan over onto the rack and tap gently all over. If the cake doesn't feel like it's falling out smoothly, lay a slightly damp kitchen towel over the pan and tap again. Cool completely before frosting, about 1 hour.

Cook the dry caramel for the frosting: Place ⅓ cup of the sugar in a heavy-bottomed pot. Cook over medium-high heat, stirring constantly, until the sugar turns to a golden-brown liquid. Remove from the heat.

Add the remaining sugar, flour, and cream: Add the remaining 2 ½ cups sugar, flour, and cream. Whisk until smooth.

Bring to a soft ball stage: Return the pan to medium-high heat and clip on a candy thermometer. Cook, stirring occasionally, until the mixture reaches 238°F on a candy thermometer, about 12 minutes. Remove from the heat.

Cool the mixture: Stir in 1 stick of the butter and vanilla. Cool, stirring occasionally, until the mixture is 110°F or cooler, about 1 hour.

Level the cakes: Use a long serrated knife to remove a thin, even layer from the top of each cake to remove any hump that occurs during baking.

Beat the cooled caramel and add the remaining butter: Use an electric hand mixer to beat the cooled caramel mixture in the pan on high speed for 10 minutes. Add the remaining 1 stick of butter and continue to beat on high speed until the butter is completely incorporated and the icing is a thick, spreadable paste.

Fill and layer the cake: Set 1 cake layer on a cake plate. Pour ¾ cup of the frosting onto the cake layer and spread it out to the edges. Invert the other cake layer onto the frosting.

Ice the cake: Pour the remaining frosting on top of the cake. Use an offset spatula to spread the frosting off the top of the cake and onto the sides.

Cool and serve the cake: Once the cake is completely frosted, cool for at least 2 hours before slicing serving.