Brown the butter by melting it in a small skillet over low heat.
Gradually bring heat up to medium-low and stir frequently.
The butter will start to look frothy and small bits of brown will start to form.
Keep stirring until the butter has a nice caramel look to it, and immediately pour the butter into a heatproof bowl. This is extremely important because it will burn if it sits in the skillet. Allow the butter to cool down before adding it to the pancake mix.
To Make Pancakes
Heat griddle or non-stick pan to medium heat and grease lightly.
In a medium bowl, stir together the cooled-down butter and the sugar.
Beat in the egg and mix in the vanilla extract.
Add in the baking powder, baking soda, and salt.
Using a spoon to stir, add in the milk and flour by adding half each at a time.
Make sure not to overbeat the mix; it will make the pancakes flat.
The batter will be a bit lumpy, but it is fine - just make sure the ingredients are mixed in.
Test the griddle or non-stick pan by spritzing water on it. If it sizzles, the pancakes are ready to be added.
Add ¼ cup of batter to the griddle or pan and let it sit for about 1 minute. Bubbles should form and pop.
Flip pancakes and heat for another 30 seconds to 1 minute longer.