Preheat the oven to 350° F (180° C.) Line 3 baking sheets with parchment paper. Add the 1/2 cup of rolled oats to a food processor and run on high until the oats become a very coarse flour.
In a medium bowl, combine the ground oats, flour, baking soda, salt, cinnamon, nutmeg, and allspice. Stir until well combined. Add the remaining 1 cup of rolled oats, and stir until they are mixed in well.
In a large bowl, with an electric mixer, cream the butter, sugar, and brown sugar until lightly fluffy. Add the egg, molasses, and vanilla, and mix until they are fully incorporated. Switch to a spoon and add the flour and oat mixture, stirring until the cookie dough is well combined.
Using a cookie scoop or your hands, shape the dough into about 20 balls of uniform size. Space them evenly on the baking sheets, giving them a lot of room to spread out. Bake the cookies at 350° F (180° C) for about 10 minutes, or until the edges are lightly browned. Allow the cookies to cool for about 30 minutes before glazing.