Heat the oil in a large, deep skillet over medium heat. Add the pork, and cook until browned evenly on all sides. Transfer to a plate. Whisk the flour into the oil in the skillet. Cook, whisking frequently, until the flour turns dark, golden brown, making a roux. Stir in the onion, celery, and bell pepper. Sprinkle with cayenne, if using. Cook, stirring occasionally, until the vegetables are tender. Return the pork to the skillet and cook 4-5 minutes. Fold in the turnips and add the broth. Bring to a boil, then reduce heat to medium-low. Cook for 45 minutes, stirring occasionally. Serve sprinkled with the chopped green onions.