In a large bowl, whisk together the eggs, granulated sugar, brown sugar, salt, and vanilla extract until sugar is dissolved; about 2 minutes. Whisk in the milk until thoroughly incorporated. Pour mixture into an ungreased 1-1/2 quart casserole dish. Sprinkle grated nutmeg over the mixture.
Place filled casserole dish inside a 13" x 9" baking dish; pour very hot water into the baking dish until water comes about half way up the side of the casserole dish. Carefully place on center rack of preheated oven and bake for about 1 hour or until a knife inserted half-way between the center and edge of the custard comes out clean. Begin checking after about 50 minutes.
Remove casserole dish to a wire rack and cool to room temperature. Serve warm or chilled. Store covered in refrigerator.