Position a rack in the center of the oven and heat the oven to 400°F.
Slicing crosswise, cut six 1/4-inch rounds from the center of one of the lemons. Repeat with the remaining lemon. Squeeze the juice from the ends of the lemons into a small bowl (you’ll need 4 tsp.) and set aside. Arrange the lemon rounds in slightly overlapping pairs on a heavy-duty rimmed baking sheet.
Season the fish with salt and pepper. Set two sage leaves on top of each fillet and then wrap each fillet with a slice of prosciutto. The prosciutto should form a belt, enclosing the leaves but leaving the fish exposed at either end. Lay one wrapped fillet on top of each pair of lemon slices. Bake until the fish is cooked through (it should flake and be opaque in the thickest part), 15 to 20 minutes, depending on the thickness of the fillets.
While the fish is cooking, melt the butter in a 10-inch skillet over medium-low heat. Add the remaining sage leaves and cook, turning once, until the leaves are crisp and the butter begins to brown, about 7 minutes. Add the 4 tsp. lemon juice to the butter and season with 1/4 tsp. salt or to taste.
Arrange each fillet (on its lemon slices) on each of six dinner plates. Pour any juices from the fish into the butter sauce in the skillet. Reheat the sauce if necessary. Spoon some of the sauce and a few sage leaves over each fillet and serve.