Preheat oven to 350°F. Grease your baking dish of choice.
Brown the pork breakfast sausage in a skillet over medium heat until fully cooked. Remove the sausage from the skillet and set aside, leaving the drippings in the pan for the gravy.
Cut the biscuit dough into about 1-inch pieces and spread them evenly across the bottom of the prepared baking dish.
Layer the cooked sausage evenly over the biscuit pieces, then sprinkle the shredded cheese over the sausage.
Pour the scrambled eggs evenly over the biscuit, sausage, and cheese layers.
To make the gravy, add enough oil to the sausage drippings in the skillet so you have about 2–3 tablespoons total fat in the pan. Whisk in the flour and cook for about 1 minute to remove the raw flour taste.
Slowly whisk in the chicken broth, scraping up any browned bits from the bottom of the pan. Bring the gravy to a gentle boil and whisk until it begins to thicken.
Whisk in the whole milk and fresh thyme. Continue cooking until the gravy is smooth and thick enough to coat the back of a spoon, about 5–10 minutes. Season with salt and plenty of black pepper, to taste.
Pour the country gravy evenly over the casserole.
Bake uncovered at 350°F until the biscuits are cooked through and the casserole is set. Start checking around 30–35 minutes, but add more time if the center still looks doughy or loose.
Let the casserole rest for a few minutes before slicing and serving.