Oil and preheat a 12" regular Dutch oven to 350 degrees F.

For the gravy, brown sausage in a the Dutch oven, breaking up and crumbling the meat as it cooks.
Once the meat is fully browned, stir in the 2 tablespoons of butter until melted.
Sprinkle the flour on top of the meat a little at a time until incorporated.
Cook, over medium heat, stirring often, for 5 minutes.
Slowly begin stirring in the milk until fully incorporated and mixture begins to bubble and thicken.
Add additional milk a little bit at a time, only if the gravy is too thick.
Remove from the heat and grind plenty of pepper directly into the gravy. Taste, add salt and adjust for seasonings. Pour gravy into a bowl.

Prepare biscuits in a bowl while the sausage is cooking, cut cold butter into the flour until you have crumbles.
Add the buttermilk and stir with a fork until a shaggy mixture forms.
Turn out onto a floured surface and quickly gather mixture to form a dough.
Fold over and pat down, turning and repeating this three times.
Quickly pat thin and cut out 8 biscuits, gathering dough scraps for the final few.
Cut each biscuit into quarters and scatter in the Dutch oven.

Top the biscuits with the shredded cheese. Whisk the eggs with 1/4 cup of milk; season lightly with salt and pepper and pour all over the top of the biscuits.
Pour the sausage gravy evenly over the top.
Bake at 350 degrees F for 40 to 45 minutes or until set. Let rest for 5 minutes, cut into squares and serve.

Cook's Notes: For more tender biscuits, try not to overhandle the biscuit dough. May also substitute an 8-count can of refrigerated biscuits and a packet of sausage gravy mix for 2 cups (such as Pioneer brand). If using canned biscuits, use the inexpensive basic biscuits and avoid the larger southern style, homestyle, jumbos or flaky varieties. If that happens to be what you have on hand, split them horizontally before quartering them so they aren't so thick.