Preheat an outdoor grill for medium heat and lightly oil the grate.

Beat the egg and onion together in a mixing bowl. Add the turkey, Italian bread crumbs, liquid smoke, Worcestershire sauce, salt, and pepper. Mix until evenly combined and form into 4 patties. Press each patty into the panko crumbs and set aside.

Cook the turkey burgers on the preheated grill until no longer pink in the center and the juices run clear, about 4 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). While the burgers are cooking, brush the mushrooms with olive oil and cook on the grill along with the Canadian bacon. Just before the turkey burgers are done, top with the grilled Canadian bacon slices and the Gouda cheese. Cook until the cheese melts.

Spread the hamburger buns with mustard and mayonnaise. Place a turkey burger onto each bottom bun and top with the portobello mushroom slices. Sandwich with the remaining bun halves and serve.