To make the cookies, preheat oven to 375 degrees.
In the bowl of a stand mixer, cream together the butter and sugars until light and fluffy. Beat in the egg and vanilla until well combined.
In a medium bowl, whisk together the flour, baking soda, cream of tartar, and salt.
Slowly add to the butter mixture and mix on low until just combined. Stir in the pecans.
Drop balls of cookie dough on a parchment lined cookie sheet 1 inch apart (they don't spread much at all) and bake for 10 - 12 minutes or until the edges are golden.
Cool on wire racks.
To make the glaze, melt the butter in a small saucepan over low heat until browned, whisking constantly, about 5 minutes.
Remove from the heat. Whisk in the powdered sugar and milk until smooth.
Use a spoon to drizzle the glaze over the cookies.