Combine sugar and water in a medium saucepan, stirring just to mix. Once combined, avoid stirring further. Cook over high heat, using a pastry brush dipped in water to wash down the sides of the pot as needed to prevent sugar crystals from forming. Continue cooking until the caramel reaches a deep amber color, then immediately remove from heat. Slowly pour in the room-temperature heavy cream while whisking rapidly — expect the mixture to bubble vigorously and boil. Add the room-temperature butter, return the pot to heat, and bring the mixture back to a boil. Reduce heat and simmer for 2 minutes, whisking constantly. If you want salted caramel, stir in sea salt now, starting with less than you think is needed. Taste the caramel after it cools slightly on a spoon and add more salt if desired. Remove from heat and allow the caramel to cool to room temperature (about 2 hours). Transfer to a container and refrigerate to thicken.

Preheat your oven to 350°F. Grease and flour three 6-inch cake pans, then line the bottoms with parchment paper. In a medium bowl, whisk together the flour, baking powder, and salt; set aside. In the bowl of a stand mixer, beat the butter and sugar on medium-high speed until light and fluffy, about 2-3 minutes. Add the eggs one at a time, fully incorporating each before adding the next. Mix in the vanilla extract and the cooled caramel until combined. Alternately add the flour mixture and milk, beginning and ending with the flour, incorporating fully after each addition (three flour additions and two milk additions). Divide the batter evenly between the prepared pans and bake for 35-40 minutes, or until a toothpick inserted in the center comes out mostly clean. Cool the cakes in their pans on wire racks for 10 minutes, then invert them onto wire racks to cool completely.

Combine egg whites and sugar in the bowl of a stand mixer and whisk to combine. Place the bowl over a pot of simmering water (1-2 inches deep), whisking constantly until the mixture is hot to the touch, no longer grainy, and reaches 160°F on a candy thermometer (about 3 minutes). Transfer the bowl back to the stand mixer and whip on medium-high speed until stiff peaks form and the mixture has cooled so the bowl is no longer warm to the touch (about 5-10 minutes). Switch to the paddle attachment and gradually add the cubed butter, mixing until smooth and creamy. Add the cooled caramel and whip until fully incorporated and smooth. If desired, add additional salt to make a salted caramel buttercream.

Trim the cake layers slightly to level them. Place one cake layer on a cake stand or serving plate. Spoon 1-2 tablespoons of the cooled caramel over the layer, spreading it evenly and pressing gently so the cake absorbs the sauce. Spread about 2/3 cup of the buttercream evenly on top. Repeat this process with the remaining layers. Apply a thin crumb coat of frosting over the entire cake and chill for 20 minutes. Then, frost the outside of the cake generously, focusing on smoothing the sides with an icing smoother. Use a cake comb on the sides, scraping off excess frosting as you go, until the sides are smooth. Using a small spoon, place dollops of caramel around the top edges of the chilled cake, allowing some to drip down the sides. Fill the center of the cake top with more caramel and spread evenly with an offset spatula. Finally, pipe rosette dollops using a 1M piping tip with the remaining frosting.