Preheat oven to 350°F (177°C). Grease and flour a 9x13 baking dish or line with parchment for easy removal.
In a large mixing bowl, beat butter and sugars until light and fluffy (2–3 minutes).
Add eggs and vanilla. Beat until smooth.
In another bowl, whisk flour, baking powder, cinnamon, nutmeg, and salt.
Add dry ingredients to wet ingredients and mix on low speed until just combined.
Add oats and mix on low until incorporated. Do not overmix.
Press dough evenly into prepared pan.
Bake for 25–35 minutes, until edges are golden and a toothpick inserted in the center comes out with moist crumbs.
Cool completely in the pan before adding icing.
Make the icing by whisking powdered sugar, salt, milk or cream, and vanilla in a bowl until glossy. Adjust thickness with more sugar or milk as needed.
Lift cooled bars from the pan using parchment if used. Cut into bars and drizzle or spread icing on top. Let set before storing.