Heat 2 tablespoons olive oil in a large skillet over medium heat. Add carrot and bell pepper; cook and stir until slightly softened, about 5 minutes. Stir in shrimp, onion, corn, and garlic; cook and stir until shrimp turn pink and are no longer translucent, about 5 minutes. Pour off any liquid, reduce the heat to low, and let simmer.

Heat remaining 1 tablespoon olive oil in a separate skillet over medium heat. Pour in beaten eggs and scramble until firm and no longer runny, about 2 to 3 minutes. Gently mix scrambled eggs into shrimp and vegetables.

Stir rice, oyster sauce, soy sauce, 3 teaspoons butter, and salt into shrimp mixture, tossing until thoroughly combined and rice and shrimp are coated with sauce.

Heat remaining 1 teaspoon butter in a small nonstick skillet over medium heat until melted and no longer foaming. Crack one whole egg into the skillet, cover, and cook until the bottom is slightly browned, the yolk is runny, and the top of the white is slightly firm, about 3 minutes. Transfer to a plate and repeat with remaining eggs, adding more butter as needed.