Preheat the oven to 350° degrees Fahrenheit.
In a mixing bowl, stir together the flour, sugar, salt, and baking soda.
In a separate mixing bowl, combine the hot water, vanilla extract, and vinegar, then add in the brown butter.
Fold together the wet and dry ingredients.
Line a 8 1/4" x 8 1/4" glass baking dish with parchment paper. Pour the cake batter into the baking dish.
Bake the cake for 25 minutes, or until a toothpick can be inserted into the middle and comes out clean.
Remove the cake from the oven, and let cool for 20 minutes.