For the cake: Melt the butter in a large shallow saucepan over medium-low heat. Cook, stirring occasionally, until the butter starts to brown and smell nutty, 10 to 12 minutes. Scrape the butter and all the browned bits into a heatproof bowl. Place the bowl over an ice bath to solidify and set aside.
Preheat the oven to 350 degrees F. Grease three 8-inch cake pans with the 1 tablespoon butter (or spray with nonstick cooking spray) and then lightly dust with flour.
Whisk together the 3 3/4 cups flour, baking powder, cinnamon, nutmeg, baking soda and salt in a medium bowl until well combined, then set the flour mixture aside.
When the brown butter has solidified, scrape it into the bowl of a stand mixer fitted with the paddle attachment. Add the brown sugar and granulated sugar and beat on high speed until very light and fluffy, about 3 minutes. Scrape down the bowl and reduce the speed to medium, then add the eggs one at a time, beating after each addition and scraping the bottom of the bowl to be sure everything is well mixed.
Add the sour cream and vanilla and beat just until well mixed, scraping down the bowl again. Add the flour mixture in three parts, alternating with the milk, beating just until incorporated. Remove the bowl from the mixer and give the batter a couple of folds with a spatula, scraping the bottom to ensure everything is evenly mixed.
Divide the batter among the three prepared cake pans and bake until golden brown and a toothpick inserted in the center comes out mostly clean with some moist crumbs sticking to it, about 25 minutes. Place the pans on a wire rack until cool enough to handle, then invert them onto the rack and unmold the cakes. Carefully turn the cakes right-side up and let cool completely on the rack.
For the assembly: Look at your cake layers. If any of them have a dome in the center, cut it off with a serrated knife so the entire top is flat. Place one cake layer on a cake plate or cardboard cake round. Spread vanilla frosting around the outside of one layer and then frost a 1-inch band around the top. Evenly spread half the dulce de leche in the center of the layer until you reach the 1-inch band and then sprinkle with half the sea salt. Add another cake layer and repeat with the frosting and remaining dulce de leche and sea salt. Add the last cake layer and frost the entire cake with a smooth and even layer of frosting. Refrigerate for about 30 minutes until the frosting is set and cold.
Remove the cake from the refrigerator and place it on a serving plate or cake stand. Using a spoon, create drips with ganache around the outside of the cake from the top edge. Pour the remaining ganache in the center of the cake and spread it with an offset spatula to cover the top. Let the ganache set completely before serving.
Spiced Chocolate Ganache:
Place the chocolate in a heatproof bowl. Stir together the cream, cinnamon, vanilla and chili powder in a small saucepan. Heat over medium heat until it just begins to bubble around the edges, then pour it over the chocolate and let sit for about 10 seconds. Stir until smooth and shiny. Let the ganache sit until it has cooled and thickened slightly, about 5 minutes.