In a medium saucepan over medium heat, heat the butter and brown sugar, stirring frequently, until the sugar has dissolved.
Cook, stirring, about 1 minute longer--the mixture will bubble but should not boil.
Set the pan aside to cool for about 10 minutes.
Meanwhile, butter an 8-inch square pan, line the pan bottom with parchment (or waxed paper), and then butter the parchment.
Stir the egg, egg yolk, and vanilla into the cooled sugar mixture.
Add the flour, salt, and nuts, stirring just until blended.
Pour the batter into the prepared pan.
Bake until the center is springy when touched (the top may still look doughy) and a toothpick inserted in the center comes out clean (it's fine if there are a few moist crumbs clinging to it), 25 to 35 minutes.
Set the pan on a rack until it's cool enough to handle.
Run a paring knife around the inside edge of the pan and then invert the pan onto a flat surface and peel off the parchment.
Flip the baked brownie back onto the rack to cool completely.
Cut into squares with a sharp knife.
This recipe can be doubled easily; use a 13x9-inch pan and increase the baking time slightly.