Preheat the oven to 350°F (180°C). Line three rimmed baking sheets with parchment paper or silicone baking mats.
In a large mixing bowl, whisk together the flour, ground cinnamon, ground nutmeg, baking soda, and salt until well combined. Stir in the quick-cooking oats.
In the bowl of a stand mixer fitted with the paddle attachment or in a large mixing bowl using a handheld mixer, beat the butter, brown sugar, and granulated sugar together until well combined, about 2 minutes.
Mix in the eggs, molasses, and vanilla extract until fully combined, making sure to stop and scrape down the sides of the bowl as needed.
Add the dry ingredients to the mixer and stir until just combined.
Using a 1 1/2 tablespoon cookie scoop, scoop the cookie dough onto the prepared baking sheets, making sure to leave 1 1/2 to 2 inches (4 to 5 cm) between each cookie.
Bake until the tops of the cookies are set and the edges are lightly browned, 10 to 12 minutes. Remove from the oven and allow to cool on the baking sheets for 5 to 10 minutes, then carefully transfer to a wire rack to cool completely.