While the dough rises, crank the oven to 400℉ and place a rack in the middle position.
Form the sausage into 12 even patties 2 1/2-inches diameter by 1/2-inch thick Place on a half-sheet pan lined with parchment paper and (when the oven hits 400℉) bake for 7 minutes. Remove the pan from the oven, flip the patties over and bake another 7 minutes. Remove from the oven and set aside to cool.
Reduce the heat to 375℉, keeping the rack in the middle position.
Crack one of the eggs into a small bowl, whisk with 1 tablespoon of water, and set aside. (This will be your egg wash to use later.) Crack the remaining 11 eggs into a large mixing bowl and whisk together with the salt, pepper, chives, and 2 tablespoons of water.
Melt the butter in a 12-inch non-stick skillet over medium heat. Add the egg and chive mixture and cook, stirring constantly with a silicone spatula until large soft curds begin to form, about 2 minutes. Kill the heat and continue to fold the curds until the eggs are barely set, another 30 seconds. Remove to a clean bowl to cool.