Use a spoon to scrape out the gills from the underside of the mushroom cap. Cut the cap into 1-inch pieces and transfer to a food processor. Add 2 Tablespoons shallot and 3 Tablespoons parsley and pulse until chopped.
Transfer the mushroom mixture to a large bowl. Add 1 ¼ pounds turkey, 2 Tablespoons olive oil, 2 teaspoons Worcestershire sauce, 1 teaspoon salt, and pepper to taste; gently mix with your hands until just combined. Divide into 4 balls, then lightly press into 4-inch-wide, 1-inch-thick patties. Put on a large plate, cover and refrigerate until firm, about 30 minutes.
Preheat a grill to medium. Brush the grates with olive oil. Grill the patties, undisturbed, until marked on the bottom, 4 to 5 minutes. Give the patties a quarter turn and cook until marked again, 3 to 4 more minutes. Flip the patties and grill until cooked through, 5 to 6 more minutes or until internal temperature reaches 165F degrees; top each with 2 slices cheese during the last 3 minutes of cooking and cover with a disposable aluminum pan to melt.
Toast the English muffins on the grill, then make a spread with mustard and mayonnaise and cayenne pepper. Start with half and half quantities of mustard and mayo, then adjust from there, increasing the mustard a bit. Add a dash to ¼ teaspoon cayenne pepper, depending on preference. Serve the patties on the English muffins; top with avocado and lettuce.