Slice the elk bottom round into eight ½-inch steaks. Using a meat mallet, tenderize both sides of each steak. Season generously with salt and pepper and set aside.
In a shallow bowl or plate, combine the Italian-seasoned breadcrumbs and Parmesan cheese. Press each steak firmly into the breadcrumb mixture, coating both sides. Shake off any excess.
Heat a large skillet over medium heat and add the oil. Once hot, carefully add four of the breaded steaks. Cook until golden brown, about 1–2 minutes per side. Remove and repeat with the remaining steaks. Set aside.
Carefully drain the oil from the skillet and return it to medium heat. Add the butter, onions, and garlic, cooking until the onions soften, about 5 minutes. Add the mushrooms and cook for an additional 5 minutes.
Sprinkle the onion and mushroom mixture with flour, stirring to coat evenly. Slowly add the beef broth, stirring constantly until the gravy thickens, about 5 minutes. Stir in the fresh thyme and reduce the heat to low.
Return the elk steaks to the skillet, nestling them into the gravy. Cover partially and simmer over low heat for 30 minutes, or until the meat is fork-tender.
Serve the elk Swiss steaks smothered in gravy with a side of baked potatoes, mashed potatoes or fries.