To make the Parmigiano Reggiano cream, combine all the ingredients in a small blender and blitz together. Season to taste and thin to a double cream consistency with hot water.
For the chicken, pull the fillets from the chicken breasts and reverse them. Season the flesh side with salt and pepper, then wrap each chicken breast tightly in clingfilm to form a tight, even cylinder. Don't tie the clingfilm but trim the excess and place the four breasts in a vacuum pouch with half the butter. Seal on full pressure and cook in a water bath at 65℃ for 30 minutes.
Remove the chicken from the pouch, retaining any liquid. Cut off the clingfilm and dry thoroughly. Heat the oil in a heavy frying pan and seal the chicken, skin side down. Add the remaining butter, sage and garlic. Baste and cook until a core temperature of 70℃ is reached. Remove the chicken to rest. Tip any fat from the pan and deglaze with the wine. Add the chicken stock. Skim any fat from the liquid from the vacuum pouch and add this to the pan. Reduce until slightly syrupy.
For the lettuce, drizzle the cut side of the lettuce with oil and season, then cook on a char grill or griddle pan. When the cut surface is well marked, turn and cover with a tin or lid to allow the lettuce to steam until tender. Alternatively transfer to a small roasting tin, cover with foil and finish in a hot oven, 180℃, for 5 minutes.
To serve, place two lettuce halves on the plate, slice the chicken on the bias into five or six slices and lay beside the lettuce. Dress the lettuce generously with the Parmigiano Reggiano cream and shavings, then drizzle some of the reduced pan juices over the chicken.