In a large bowl, whisk together the rolled oats, quick oats, gluten-free flour, baking soda, salt, cinnamon, and nutmeg.
In a stand mixer fitted with the paddle attachment, beat the softened butter until creamy. Add the brown sugar and white sugar and beat for 2 minutes.
Add the eggs, molasses, and vanilla. Mix until combined.
Add the dry ingredients and mix on low speed until just combined.
Cover and refrigerate the dough for 45 minutes.
Preheat the oven to 350°F and line baking sheets with parchment paper.
In a medium bowl, whisk together the powdered sugar, milk or cream, and vanilla until smooth.
Scoop the dough into 2-tablespoon portions and place on the prepared baking sheets, spacing about 2 inches apart.
Bake for 10–12 minutes, until the edges are lightly golden and the centers are just set.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack.
While the cookies are slightly warm, dip the tops into the icing. Return to the rack and allow the icing to set before serving.