Melt the butter in a small saucepan over medium-low heat. Whisk in the flour, and stir until the mixture becomes paste-like and light golden brown, about 3 minutes. Gradually whisk the milk into the flour mixture, and cook over low heat. Cook and stir until the mixture is thick and smooth, about 10 minutes. Stir in the Parmesan cheese and hot sauce; season to taste with salt and pepper. Keep warm over low heat.

Meanwhile, warm the biscuits in a toaster oven and keep warm. Heat the vegetable oil in a large skillet over medium heat. Place the andouille pieces into the skillet cut-side-down. Cook until golden brown, then turn over and cook until browned on the skin side; remove and keep warm. Reduce the heat to medium-low. Crack four of the eggs into the hot pan, and cook until the egg whites have firmed about halfway through, about 1 minute. Gently flip the eggs over, and cook 10 to 20 seconds more until the other side has just firmed on the outside. Repeat with the remaining eggs.

To assemble, place two biscuit halves onto each plate, cut-side-up. Divide the andouille sausage among the biscuits, then top each biscuit with an egg. Finally, spoon some of the sauce over each egg and serve.