Heat a medium sauté pan over medium heat, add the butter and allow to melt.
Continue to cook, stirring constantly, until the butter turns deep golden brown and small brown bits form at the bottom, about 8 minutes. Immediately pour the butter into a small bowl to stop the cooking process. Set aside and let cool to room temperature.
In a medium mixing bowl whisk together the flour, baking powder, baking soda and salt.
In another large mixing bowl whisk together 2 tablespoons of the brown butter, egg, buttermilk, brown sugar and vanilla until combined. Mix the dry ingredients into the wet until just combined.
Heat a medium non-stick skillet or griddle over medium-low heat. Add about 2 tablespoons of the remaining brown butter and heat.
Add about ½ cup of the pancake mixture to the pan and cook until bubbles appear and the underside is golden brown, about 2 minutes.
Use a spatula to carefully flip the pancakes and continue to cook until the pancakes are cooked through and golden brown, about an additional 2 minutes. Set aside and repeat with the remaining batter and brown butter.
Serve hot, topped with maple syrup and flake sea salt.