First you need to make the crust. Preheat the oven to 350. Line a 8x8 pan with foil, set aside. In a medium bowl mix together your melted butter, sugar and vanilla. Add the flour and salt and stir well. Press mixture into the bottom of your prepared pan. Bake for 17 minutes or until golden brown. Let cool a bit on a wire rack.
While your crust is baking you need to brown your butter. By the way, brown butter is sooooo good! It tastes a little nutty. I love it! So, to brown it you'll need to cook your butter in a small saucepan. A lighter colored one is best so you can see the color of the butter change. Cook it over medium heat stirring often and watching it closely. It'll take about 6 minutes to brown. Once brown, immediately pour it into a glass measuring cup to cool slightly.
Now you need to whisk your eggs, sugar and salt in a medium bowl to blend. Add flour and vanilla and whisk until smooth. Slowly whisk in your browned butter into your egg mixture. Whisk until smooth. It should look something like this....
Check out that shortbread crust and speckles of browned butter. Heaven.