Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high heat. Add garlic and cook 30 seconds. Season shrimp with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper. Add shrimp to skillet and cook 2 minutes per side. Remove to a plate and keep warm.
Add chard leaves to skillet and season with the remaining 1/2 teaspoon of salt, 1/8 teaspoon pepper and the oregano. Reduce heat to medium-low. Cook 4 to 5 minutes until tender. Add shrimp back into skillet. Cover and set aside.
Meanwhile, cook pasta following package directions, about 10 minutes. Drain, reserving 1/2 cup cooking water. Toss pasta with shrimp mixture and remaining 2 tablespoons oil. Add enough of the pasta water to create a sauce. To serve, transfer to a large bowl and sprinkle with cheese.