In a medium bowl, use your hands to mash the hot Italian sausage and ground beef together until evenly combined. Try not to over mix. Form into to equal patties, each 1/4 lb. and about 1/2" thick, and set aside.
Place a cast iron skillet over medium heat. Slice English muffins in half, and butter the inside halves of both muffins.
Place the English muffins, butter side down, into the cast iron skillet. Allow to cook 2-3 minutes, until browned. Flip. Cook another 1-2 minutes on the exterior of the muffins. Set aside and cover with foil to keep warm.
In the same skillet, add the sausage patties. Cook until browned with a crispy exterior, and cooked through; about 4-5 minutes per side, 8-10 total minutes. Add 1/4 cup of shredded cheese directly onto each sausage patty, and cover the skillet with a lid. Allow the cheese to melt; roughly 30 seconds. Remove sausage patties from skillet and set aside, also covering to keep warm.
In the same skillet, using the excess fat from the sausage patties as oil, crack in two eggs. Cook as desired (over easy, sunny side up, etc.) At least fully cook the whites. Season with salt and pepper on top of each egg.
Build the sandwiches with the bottom half of the English muffin, then sausage patty with cheese, and top with the fried egg. Add hot sauce and finely chopped chives, if desired.
Serve hot. Enjoy!