Add the butter, granulated sugar and brown sugar to the bowl of a stand mixer. Mix together on high speed until it looks pale yellow, 4 to 5 minutes. Reduce the mixing speed to medium. Add the vanilla extract and molasses Add the egg and mix until the ingredients are fully incorporated. The dough should be a smooth, consistent texture. Scrape down the sides of the bowl.
In a small bowl, whisk together the flour, salt and baking soda and set to the side.
Reduce the mixing speed to low and add the dry ingredients slowly. Mix until all of the dry ingredients are saturated. Scrape down the sides and bottom of the bowl, folding in any unincorporated ingredients.
After the dough is well mixed, remove the dough from the mixer and place on a piece of parchment paper. Roll the parchment to create a log approximately 1 1/2 inches in diameter (about 12 inches long). Wrap the log in plastic wrap and place in your refrigerator for at least 2 hours.
Preheat the oven to 350 degrees F.
Remove the chilled, firm log from the refrigerator. Remove the parchment and, using a sharp knife, slice the cookie dough in 1/4-inch circles. Place the cookies on a parchment-lined baking sheet. The cookies will expand in the oven, so leave about 1 cookie width of space between them on the sheet (about 12 cookies per sheet). Work quickly, as these cookies should go into the oven while the dough is still cold.
Bake for 8 to 10 minutes. The cookies are ready when they are just beginning to brown and let off a fragrant brown butter smell. Allow the cookies to cool fully before serving.