Pat the venison steak dry with a paper towel and season both sides generously with salt and pepper.
Heat a large cast-iron skillet over medium-high heat. Add the oil to the skillet, carefully swirl it around to coat the pan, and then add the venison steak.
Let steak sizzle for about 2 minutes, then use tongs to flip it over and cook about 2 more minutes over medium heat. Press down gently to ensure even contact between steak and pan. When the steak has developed a dark brown crust on both sides, add 2 tablespoons of the butter.
After another one minute, use an instant-read thermometer to insert into side of steak to check the internal temperature. For medium-rare meat, 125 to 130 degrees is ideal. Keep checking the steak with the thermometer every 90 seconds until you reach the proper temperature.
Once you've reached a perfect medium-rare temperature, remove the steak and transfer it to a cutting board and let it rest for 5 to 10 minutes while you fry the eggs.
Keep the heat at medium and add the remaining 1 tablespoon butter to the cast iron skillet. Crack the eggs into the skillet, keeping them separate, and cook until done to your preference, about 4 minutes for soft sunny–side up. Season with salt and pepper.
Serve the venison steak and eggs with hashbrowns or toast.