Whisk eggs, milk, seasoned salt, salt, and black pepper together in a bowl.

Heat oil in a large nonstick skillet over medium-high heat. Add jalapeño and sauté until slightly softened, 2 to 3 minutes. Add ham and cook until heated through, about 1 minute.

Pour egg mixture into the skillet. Cook and stir until eggs are set but not dry, 3 to 5 minutes. Sprinkle 1/2 of the Cheddar over eggs; cook and stir until cheese is melted. Transfer eggs to a plate and sprinkle remaining Cheddar over the top.