1. Preheat oven to 400 degrees. Brush four 8-ounce ramekins or other small ovenproof dishes with oil. Prepare grits according to package instructions. Stir in arugula and 2 tablespoons cheddar, and cook until cheddar melts, about 30 seconds. Season with salt and pepper.
2. Divide grits among dishes. Make a well in each, then crack an egg into each well; season. Bake on a rimmed baking sheet, until whites are set but yolks are still runny, about 20 minutes, rotating sheet halfway through. To serve, top grits with remaining 2 tablespoons cheddar and a sprinkle of pepper.