Heat oil in a heavy oven-proof skillet over medium-high heat. Add ham and cook until browned all over, about 3 minutes per side. Remove from heat and cut ham into 4 serving pieces.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add cauliflower and broccoli, cover, and steam until tender, 2 to 6 minutes.
Squeeze the pouch of wild rice mix to separate the rice. Combine rice and 2 tablespoons water in the skillet over medium heat. Cook and stir until rice is thoroughly heated, about 2 minutes. Spread rice evenly over the skillet. Cover with the cauliflower and broccoli. Layer the ham steak, Cheddar cheese, and mushrooms on top.
Blend celery soup, sour cream, mustard, and curry powder together in a bowl. Pour over the cheese and mushroom layers. Sprinkle with Parmesan cheese.
Bake in the preheated oven until ham steaks are no longer pink in the center, about 1 hour. An instant-read thermometer inserted into the centers should read 145 degrees F (63 degrees C).