Heat the broiler, setting its rack 4 inches from the heat.
Season cod with salt and pepper. Wrap 1 slice of prosciutto around each cod fillet. Rub with 1 teaspoon of oil and place seam side down on a broiler-proof baking sheet.
Broil until the prosciutto is browned and fish is opaque throughout. This should take 8 to 12 minutes, depending on the thickness of your fish. (I cooked ours about 10 minutes. The fish will continue to cook a bit more once you pull it from the broiler.)
While the cod is broiling, cook the spinach. In a large skillet, heat the remaining tablespoon of olive oil over medium heat. Fill the skillet with as much spinach as it will hold. Season with salt and pepper. Cook, tossing and adding more spinach to the skillet, when there is space, until it is all wilted and tender. This will take about 5 minutes.
Sprinkle spinach with lemon juice and toss to combine.