In a medium bowl, combine the oatmeal, butter, and boiling water, and stir to mix them together a bit. Set aside for 20 to 30 minutes.
Heat the oven to 350°F (175°C) and generously butter and flour two 9-inch round cake pans or one 13-by-9-inch pan.
In a medium bowl, combine the flour, baking soda, salt, and nutmeg, and stir with a fork to mix everything well. In a large bowl, combine the granulated and brown sugars, eggs, and vanilla and beat with a mixer on medium speed for about 2 minutes, until thick and light colored, stopping to scrape down the bowl when neccessary.
Stir the flour mixture into the egg mixture in 2 batches, beating just long enough to make the flour disappear after each addition. Mix in the oatmeal, stirring and folding to combine everything into a nubbly but well-mixed batter.
Scrape the batter into the prepared pan(s) and bake for 25 to 30 minutes, until the cake is golden brown, springs back when touched lightly in the center, and begins pulling away from the sides of the pans.
Cool the cake in the pan(s) on wire racks or folded kitchen towels for 10 minutes.If using a couple round cake pans, carefully turn the cakes onto wire racks, turn them over again so they are top side up, and let cool completely.If using a rectangular pan, let the cake cool completely in the pan.