Pat the steak dry with paper towels, season both sides of the steak generously with salt and pepper. Press salt and pepper into meat so it adheres.
In a large skillet over medium heat, arrange the mushrooms in a single layer. Cook, undisturbed, until beginning to brown, about 4 minutes. Sprinkle the mushrooms all over with kosher salt. Cook, stirring, until the mushrooms begin to release liquid, about 4 more minutes. Transfer cooked mushrooms to a plate and save for later.
Pour 1 Tablespoon of canola oil into a skillet and swirl to coat. Using tongs, gently lay deer steak in skillet and cook over medium-high heat until underside is nicely browned, and a crust has formed, about 3 minutes. Once or twice, lift up steak and reposition over hotter areas of the skillet to encourage even cooking.
Flip the deer steak over, reduce heat to medium, and cook until reverse side is equally browned and steak is medium-rare (an instant-read thermometer inserted into the thickest part will register 125 degrees F), another 3 or 4 minutes. Transfer steak to a cutting board to rest for 10 minutes while you make the mushroom sauce.
Meanwhile, in the same skillet melt butter over medium-high heat and stir in reserved mushrooms with 1 Tablespoon of minced garlic. Sauté about 3 minutes until mushrooms are golden brown.
Stir in beef broth, Dijon mustard, Worcestershire sauce, heavy cream, ½ teaspoon kosher salt, and ¼ teaspoon black pepper. Continue to stir periodically for about 5 minutes until liquid is reduced and thickened. Taste and add more salt and pepper if needed.
Thinly slice cooked deer steak with a sharp knife, cutting away from your body and with the top edge of the knife leaning toward your body. Make sure to slice across the grain of the meat. Serve mushroom sauce over steak and top with additional cracked black pepper if desired.