Preheat oven to 350°F. Grease and line the sides and bottom of a 9 X 13-inch baking pan. Set aside.
In a bowl with a hand mixer (or a stand mixer with a paddle attachment), mix the butter until creamy. Add the granulated and light brown sugar and mix until fluffy and combined.
With the mixer on low, add the eggs, one at a time. Add the vanilla and mix to combine.
Sift in the all-purpose flour, cream of tartar, baking soda, and salt. Mix until the flour begins to incorporate. Switch to a rubber spatula when the dough gets too thick.
Tip the dough into the prepared pan and begin to smoosh and spread it out. Use an offset spatula to spread until even.
In a small bowl, combine the 3 tablespoons sugar and 1 ½ teaspoon cinnamon. Sprinkle over the top of the cookie dough.
Bake for 20 to 22 minutes. When done, the edges should be set but the center still soft.
Cool the cookie bars in their pan over a wire rack before frosting.
To Make the Cinnamon Icing
Place the butter in a bowl and mix with a hand mixer (or in stand mixer with the paddle attachment) until creamy.
Add the confectioners' sugar, milk, vanilla, and cinnamon. Mix on low until incorporated.
Turn the mixer up to medium and mix until smooth and fluffy.
Use an offset spatula or the back of a spoon to spread and swirl the icing on top of the cooled cookie bars.
Sprinkle with cinnamon and slice into 18 bars before serving.
Notes
Store the frosted cookie bars in an air-tight container at room temperature for up to 3 days.Unfrosted snickerdoodle bars may be frozen. Wrap the cookie bars in plastic then slip into a zip-top bag. Cookie bars can be frozen for up the three months.