Preheat oven to 350°F. Line an 8 x 8 inch baking pan with parchment paper. Set aside.
Melt the butter and transfer to a medium size bowl. Set aside to cool for about 10 minutes. Meanwhile, pulse the oats in a food processor until oats are cut into smaller bits. Do not pulse so much that the oats become “oat flour”.
In separate bowl, whisk together the pulsed oats, flour, baking soda, cinnamon and salt. Set aside.
To the melted butter, add the granulated and brown sugar. Mix on low speed (or use a whisk) for 1 minute. The mixture will resemble wet sand.
Add the egg, followed by the vanilla and molasses. Mix to combine well.
Add the dry ingredients to the butter and sugar mixture. Use a large spatula or wooden spoon to fold the mixture together just until the flour is incorporated. Do not over mix.
Transfer the dough to the prepared baking pan. Bake for 20 to 24 minutes or until the edges are set and the top is a rich, golden brown. Do not overbake or the cookie bars will be dry.
Remove the bars from the oven and cool in the pan on a wire cooling rack. Cool the bars for 1 to 2 hours until room temperature. Be sure the bars are cool to the touch before adding the icing.
In a medium bowl, whisk together the powdered sugar, milk and vanilla until thick but smooth. The icing should not be too runny. If too thin, add more powdered sugar one tablespoon at a time.
Once the bars are cool to the touch, use a spoon to spread the icing over the tops of the bars, leaving small bits of the cookie bars peeking through.
Allow the icing to set and then slice and serve.
