For the sausage gravy: In a large skillet, cook the sausage over medium heat, breaking it up with a wooden spoon into crumbles until cooked through, 8 to 10 minutes. Stir in the flour and cook about 2 minutes over medium-low heat. Whisk in the milk and cook until the sauce begins to thicken. Season with salt and pepper.
For the eggs: Cut each Biscuit in half and place the bottom halves on a baking sheet. Place a slice of American cheese on each bottom half, then bake just until the cheese is melted, 5 to 8 minutes. Top each with a hash brown patty, a generous spoonful of sausage gravy, one egg and the chives. Top each with the remaining biscuit half and serve warm.
Preheat the oven to 400 degrees F. Grease a baking sheet.
In a large mixing bowl, combine the flour and the chilled, cubed butter. Using a pastry blender, cut the butter into the flour until it resembles a coarse meal. Stir in the baking powder, salt, sugar, and baking soda. Form a well in the center of the flour and pour in the buttermilk. Mix with a fork until all of the ingredients are incorporated and the dough begins to shape into a ball. (The dough will be slightly dry.)
Dust a clean work surface with flour, then coat your palms and rub some flour on a rolling pin. Turn out the dough on the work surface. Knead the dough for 1 to 2 minutes, folding it over onto itself each time. Roll the dough to about 1/2-inch thick. Dip a 3-inch biscuit cutter or the rim of a 3-inch-wide glass in flour, then cut out the biscuits. Reshape the leftover dough into a ball, roll it out again, and cut out more biscuits. Repeat this process until the dough is gone.
Place the biscuits on the baking sheet and bake until the tops are golden brown, 10 to 12 minutes. While the biscuits are still hot, brush some of the remaining butter on each biscuit.