Preheat the oven to 350°F. Spray a 13×9-inch baking pan with baking spray.
In a large bowl, use an electric mixer to beat 3/4 cup (1 1/2 sticks) softened butter, 1 3/4 cups sugar, 3/4 teaspoon salt, 2 1/2 teaspoons baking powder, and 2 teaspoons vanilla extract until light and fluffy, about 5 minutes.
Add the 4 large eggs and 2 egg yolks one at a time, beating well after each addition and scraping down the bowl as needed.
Add 1 1/2 cups buttermilk and mix on low speed just until incorporated.
Add 2 3/4 cups all-purpose flour and mix on low speed just until combined; scrape the sides and bottom of the bowl to ensure an even batter. Do not overmix.
Pour the batter into the prepared 13×9-inch pan and smooth the top. Bake for about 35 minutes, or until the cake springs back when pressed lightly in the center (or a toothpick inserted into the center comes out with a few moist crumbs).
Remove the cake from the oven and set it on a wire rack. While the cake cools slightly (it should be warm but not piping hot), make the icing.
For the caramel icing: melt 6 tablespoons (3/4 stick) butter in a small saucepan over low heat. Turn the heat off and whisk in 1 cup packed light brown sugar and 1/4 cup milk until the sugar is dissolved and the mixture is smooth.
Stir in 1 3/4 cups powdered sugar until the icing is smooth. If the icing is very runny, let it cool a couple of minutes until it thickens slightly, but do not let it set.
Slowly pour the icing over the warm cake in the pan and spread gently if needed. Let the cake cool completely so the icing sets before slicing and serving.