Pour EVOO, sherry, Worcestershire and pepper into a dish and mix with your fingers or a whisk. Turn each mushroom disk a few times in the sauce as you add them to the dish; season with sea salt. Let marinate 30-45 minutes, turning occasionally.
Preheat a grill or grill pan over medium-high heat. Grill the mushrooms, turning once, until well marked, 4-6 minutes.
Make 4 stacks in this way: mushroom disk, cheese slice, mushroom and cheese. Place on the grill until the cheese melts, 1 minute. Transfer the stacks to the roll bottoms and top with the tomato slices; season with salt and pepper. Add the greens. Slather the bun tops with the mustard and set in place. Serve with the chips.