In a medium bowl, combine flour, baking soda, and salt. Set aside.
In the bowl of a stand mixer, beat together the butter and sugar until light and fluffy (several minutes). Add the egg and vanilla and beat until combined.
Mix in the dry ingredients until just combined. Stir in the chopped pecans. Cover the dough and refrigerate for at least one hour.
Once the dough is chilled, preheat your oven to 350 degrees F.
Shape the dough into 1-inch balls and place two inches apart on a cookie sheet lined with parchment paper. Cookies will flatten out while cooking if the dough isn't VERY cold. You may also choose to create your dough balls and freeze 5 minutes prior to baking. These also cook best on a metal baking sheet as opposed to stone. Bake for 10 minutes or until set. The cookies should just start to lightly brown. Allow to cool on baking sheet for 5 minutes before transferring to cooling rack.
Meanwhile, prepare your frosting. In a pot over medium heat, combine the brown sugar and milk. Bring to a boil and boil for about 3-4 minutes, stirring constantly. Remove from heat and stir in the butter and salt.
Then, stir in the confectioner's sugar. Whisk or beat with an electric mixer until smooth. The frosting will firm up as it cools.
Add about 1 tablespoon of frosting to each cookie. You must work quickly while the frosting is hot as it will immediately set. If it starts to not look so pretty when you add it to the cookies, heat it slightly to bring it back to the consistency you like. Top with a pecan half.
Store the cookies in an air tight container once fully cooled and the frosting has set.