Preheat your oven to 350°F. Grease a standard 12-cup muffin tin (or more) and set it aside.
In a large bowl, place the flour, xanthan gum, cornmeal, baking powder, and salt, and whisk to combine well.
Add the shortening, mix to combine and use clean hands to break the shortening up into small crumbs combined well with the dry ingredients.
Add all but about 3 to 4 tablespoons of the shredded cheese, and mix to evenly distribute the cheese throughout the flour mix.
Create a well in the center, add the buttermilk, sour cream, and egg, and mix to combine. The batter will be very thick.
To make full-sized breakfast corn muffins, divide the mixture evenly among the prepared wells of the muffin tin (filling each well up to 3/4 of the way).
Using wet fingers, press the batter down into an even layer in each well. To make corn muffins with eggs on top, place only enough muffin batter in the prepared wells of the muffin tin to fill the well about 1/4 of the way full. Using wet fingers, press the batter into the bottom of the well and halfway up the sides.
Crack a whole egg in the center of the well, covering the muffin batter.
Sprinkle the reserved shredded cheese lightly in the center of the batter (or egg) in each muffin well. Place in the center of the preheated oven and bake for about 22 minutes.
The egg-topped muffins are done when the egg white is cooked completely (ideally, the yolk is still slightly soft in the very center).
The standard muffins are done when a toothpick inserted in the center comes out mostly clean. Do not overbake.
Remove from the oven and, using a toothpick, loosen each muffin in the well so that it rotates freely. Allow the muffins to cool in the tin for about 5 minutes before transferring to a wire rack to cool completely.
Once the muffins are cool, you can place them in a sealed plastic or glass container and store them at room temperature for up to 2 days (as long as your kitchen isn’t very warm).
For longer storage, freeze in a well-sealed freezer-safe container and defrost in the microwave or toaster oven.