In a large bowl, whisk the oats with the flours, baking soda, salt, cinnamon and nutmeg.
In the bowl of a stand mixer fitted with the paddle, beat the butter with both sugars and the vanilla at medium speed until light and fluffy, about 3 minutes.
Add the eggs 1 at a time and beat until incorporated.
Add the dry ingredients and beat at low speed until just combined.
Cover the bowl with plastic wrap and refrigerate the cookie dough until chilled, at least 1 hour.
Line 2 large rimmed baking sheets with parchment paper.
Working in 2 batches, using a 1-ounce ice cream scoop or two tablespoons, scoop the dough onto the prepared sheets, spaced at least 2 inches apart.
Flatten the cookies slightly and sprinkle each with a pinch of sea salt.
Bake for 10 minutes, until golden brown at the edges; rotate the baking sheets from top to bottom and front to back halfway through baking.
Let the cookies cool on the baking sheets for 5 minutes, then transfer to a rack to cool completely.