Bring 3 cups water to a boil in a medium saucepan. Add the grits, reduce the heat to medium and cook, stirring occasionally, until thickened, about 5 minutes. Remove from the heat. Stir in the cheese and season with salt and black pepper; cover to keep warm.
Crack the eggs into a large bowl. Heat the vegetable oil in a large nonstick skillet over medium-high heat. Add the okra, scallion whites, bell pepper, cayenne, 1/4 teaspoon salt, and black pepper to taste; cook, stirring, until the vegetables start to soften, 2 to 4 minutes.
Carefully slide the eggs on top of the okra mixture, keeping the yolks intact. Reduce the heat to medium; cover and cook, rotating the skillet occasionally, until the eggs are just set, about 6 minutes.
Serve the grits with the vegetables and eggs. Top with the scallion greens.