Southwest Hash Brown Omelette on the Griddle

Name: Southwest Hash Brown Omelette on the Griddle
Ingredients: 2 links chorizo sausage (like Johnsonville Chorizo Sausage); ¼ onion (small diced); ½ bell pepper (small diced); all-purpose seasoning (or salt and pepper); ½ tablespoon high-heat cooking oil (like avocado oil); 1 tablespoon butter; 20 oz bag shredded hash browns (see notes); ½ roma tomato (diced); 1-2 tablespoons pickled jalapenos (diced); 6 eggs; 4 slices pepper jack cheese; ⅔ cup shredded cheddar cheese; hot sauce, Cotija cheese, and more shredded cheddar cheese (for garnish)