Preheat oven to 325 degrees. Line a 9x13 inch pan with aluminum foil and lightly spray with cooking spray.
In a medium sized saucepan add the butter. Cook over medium heat and whisk the butter to brown. It will become frothy and start to have a nutty aroma. Then it will turn brown. It takes about 3 minutes. Remove from heat.
In a large mixing bowl, combine dark brown sugar, sugar, vanilla, brown butter and eggs. Beat until incorporated.
Add flour and salt and mix until just combined. Do not over mix. The dough will be thick. Spread evenly into the bottom of your prepared 9x13 inch baking pan. Bake for 22-25 minutes or until the edges are lightly brown. Remove from oven and allow to cool on a wire rack completely.
To make the salted caramel buttercream:
In a medium sauce pan add the butter and brown sugar. Bring to a boil. Stir in the milk and bring back to a boil. Remove from heat and stir in powdered sugar. Frost the cooled bars ad sprinkle lightly with sea salt.