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 the shallot and parsley and pulse until chopped.
Transfer the mushroom mixture to a large bowl. Add the turkey, olive oil, 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, 4 to 5 more minutes. Flip the patties and grill until cooked through, 6 to 7 more minutes; 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 spread with mustard and mayonnaise. Serve the patties on the English muffins; top with avocado.