For the bottom layer: In a large pan on medium-high heat, add the oil, sausage and chorizo. Using a wooden spatula or, even better, a wired and wavy potato masher, break up the chorizo and sausage until they are homogenous and resemble ground beef in its tiniest form. Once broken down, continue to cook until the sausage is cooked through and orange fat is rendered from the chorizo, 10 to 12 minutes. Using a slotted spoon to drain off excess fat, transfer the mixture to an 8-by-8-inch casserole dish.
For the middle layer: In the same pan with the remaining fat from the meat, quickly add and saute the onions and peppers with a pinch of salt and pepper until tender and the moisture from being frozen is cooked out, 5 to 8 minutes. As before, use a slotted spoon to transfer the pepper/onion mixture to the casserole dish, draining off excess fat, and evenly spread this mixture over the top of the meat layer.
Next, evenly layer the mozzarella cheese over the vegetables. Then use the American cheese slices to create another layer, pressing down once done to make everything level and compact.
For the top layer: In a large bowl, add the hash browns, butter, steak rub and remaining fat from the pan. Season generously with salt and pepper and toss to coat. Sprinkle this layer on top of the cheese and gently press down, but not too tight so steam can escape.
Bake, uncovered, until the top is golden brown, 25 to 30 minutes. Serve with a shake of hot sauce or sunny-side up eggs over the top.