Preheat the oven to 500 degrees F (260 degrees C). Spray a large sheet pan with cooking spray.
Arrange potatoes, carrots, and onion on the sheet pan in a single layer. Drizzle with olive oil and sprinkle 1 tablespoon Italian seasoning on top.
Mix butter and 1 teaspoon Italian seasoning together until well blended.
Rinse hens well, inside and out, under cool running water. Remove bag from inside the cavity. Pat the hens dry with paper towels. Season the cavity with salt and pepper. Make a pocket between the skin and the meat with your fingers, being careful not to rip the skin.
Divide butter mixture between the hens and place it between the breast meat and skin. Place hens on top of the potato and onion mixture. Season the outsides with salt, pepper, and remaining Italian seasoning. Place lemon inside the cavity of the hen. Pour chicken broth into the bottom of the sheet pan.
Roast in the preheated oven until hens are nicely browned, 35 to 45 minutes. Tent the hens with foil during the last few minutes of cooking to prevent overbrowning, if needed. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C). Remove from the oven, cover with a doubled sheet of aluminum foil, and allow to rest in a warm area for 10 minutes before slicing.
Meanwhile, drain about 1/2 cup of the pan drippings from beneath the vegetables into a small saucepan to start the gravy. Bring to a boil. Mix flour and cold water together and add to the pan drippings, whisking until desired consistency is reached. Serve gravy alongside Cornish hens.