Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain and immediately return spaghetti to the pot.

Place shrimp in a large pan over medium heat. Add 1 tablespoon olive oil; season with salt and pepper. Cook and stir shrimp until mostly opaque, about 5 minutes. Add shrimp to pasta; cover to keep warm.

Heat remaining 1 tablespoon olive oil in the same pan. Add onion; cook and stir until translucent, about 5 minutes. Add garlic and cook until fragrant, about 1 minute. Add milk and half-and-half; cook until heated through, about 3 minutes.

Pour 1/4 cup of the milk mixture into a bowl; add eggs and stir until warmed through. Pour egg mixture over shrimp and pasta and stir in remaining milk mixture.