In a large cast-iron skillet, heat oil. Add hash browns and cook undisturbed for 10 minutes, then flip and cook 5 minutes more, until golden and crispy. Add onions and bell pepper and season with salt and pepper. Cook until tender, 5 minutes more. Add garlic and sliced jalapeño and cook until fragrant, 1 minute.

Add cheese to hash browns and let melt, 2 minutes. Stir together so hash browns are fully cheesy.

Make four nests for eggs and crack eggs in each. Season eggs with salt and pepper. Cover and cook until whites are set and yolks still slightly runny, 8 minutes.

Meanwhile, make breakfast gravy: Cook sausage in a medium skillet over medium heat until browned all over. Sprinkle flour over the sausage and cook 1 minute. Pour over milk and bring mixture to a boil. Reduce heat and simmer until very thick, about 5 minutes. Season with salt, pepper, and cayenne. Remove from heat.

Serve hash drizzled with breakfast gravy and garnished with chives.