Preheat the oven to 325°F. Generously grease and flour a 10-inch Bundt pan, making sure to thoroughly coat all the grooves.
Add the cake mix and brown sugar to a large mixing bowl. Whisk briefly to combine and break up any lumps.
Add the 2 whole eggs, egg yolk, sour cream, vegetable oil, evaporated milk, caramel sauce, vanilla extract, and salt.
Using an electric mixer on low speed, mix just until all the ingredients are moistened and mostly incorporated. Stop and thoroughly scrape the sides and bottom of the bowl.
Mix on low for just 20-30 seconds longer, until the batter is smooth and combined. Do not overmix.
Spoon the batter evenly into the prepared Bundt pan and smooth the top. Gently tap the pan on the counter a couple of times to remove any large air pockets.
Bake for 48-55 minutes, beginning to check the cake around 48 minutes. The cake is done when a toothpick inserted into the thickest portion comes out clean or with a few moist crumbs attached.
Cool the cake in the Bundt pan for 10-15 minutes. Carefully invert it onto a cooling rack and allow it to cool completely before frosting.
To make the frosting, beat the softened cream cheese and butter until completely smooth and creamy. Gradually add the powdered sugar and beat until incorporated. Mix in the vanilla extract.
Add evaporated milk 1 teaspoon at a time, if needed, until the frosting is thick, smooth, and pipeable.
Transfer the frosting to a piping bag fitted with a large round tip. Pipe thick ribbons of cream cheese frosting from the outside edge of the Bundt cake up toward the center.
Finish with a light sprinkle of flaky sea salt. Now this step is optional, but it really gives the best flavor!