Mix butter and brown sugar in a large bowl with a hand mixer or electric mixer fitted with a paddle attachment on high speed until creamy. Add vanilla extract and mix well.
Mix the butter and flour in the mixer bowl without using the mixer stand, if using.
Then, take the bowl back to the mixer stand and combine the dry ingredients until they are well combined. It will be a delicate dough.
Mix the milk into the dough slowly until it comes together. Beat the dough until it becomes smooth.
Turn out the dough onto a piece of plastic wrap. Make a log that is approximately 1 1/2 x 2 1/2 inches in size by wrapping firmly. Keep it in the fridge for at least three hours or all day.
Preheat the oven to 350 degrees Fahrenheit. Line a cookie sheet with silicone baking mats or parchment paper.
Cut the dough log into 1/4-inch-thick pieces. It is important to leave enough space around the cookies since they will expand as they bake.
It takes 12 to 15 minutes for the cookies to turn golden brown. Allow the cookies to sit on the cookie sheet for about 6 minutes before transferring them to a wire rack to cool.