In a large saucepan of boiling salted water, blanch lettuce for 3 minutes and refresh. Drain gently, squeeze dry. Slice each head in half lengthwise. Sprinkle with salt and pepper. Fold in half crosswise and shape with your hands to make fat triangles.
Preheat oven to 350 degrees F and set rack in lower third of oven.
In a casserole, cook the scallion and carrot in the butter until tender but not browned. Arrange the lettuce in the casserole, and spoon the vegetables over the lettuce.
Pour in enough liquid barely to cover the lettuce. Add the thyme and bay leaf. Bring to a simmer on top of the stove. Place the buttered paper over the lettuce to cover the casserole.
Place casserole in center of preheated oven. Regulate heat so lettuce simmers slowly for 1 hour. Remove the lettuce to a serving dish and keep it warm. Quickly boil down the braising liquid until it has reduced to a syrup (about 1/2 cup).
Sprinkle with parsley and serve.