Pat the steaks dry with paper towels and then liberally sprinkle salt all over them. Place on a baking sheet and let rest for 1 hour, or overnight in the refrigerator on a plate loosely covered with a paper towel.
Gently rinse the steaks with running water, helping to remove all the salt. Use paper towels to completely dry off the steaks. Season the steaks with a couple of pinches of salt and pepper. Set aside.
Turn your broiler on HIGH.
Toss the roughly chopped tomatoes with the extra-virgin olive oil, rosemary, chives, ½ tsp salt, and ¼ tsp black pepper. Place on a baking sheet lined with foil and place 6 inches under the broiler. Roast for 1 to 2 minutes, and then stir the tomatoes around with a wooden spoon. Roast for another couple minutes, until soft and starting to char. Remove and loosely tent with a piece of foil. Keep your eyes on them in the oven, don't let them completely burn!
Heat ¼ cup of the canola oil in a large cast-iron skillet over medium-high heat until shimmering and just starting to smoke.
Add one steak, and sear, turning frequently with a spatula and tongs, until browned and crust has started to form on the outside of the steak, about 3 to 4 minutes Add the rosemary sprigs and garlic cloves, and butter.
With oven mitt on, carefully tilt the skillet to one side and use a large spoon to spoon the butter/oil all over the steak, flipping and continuing to spoon. Repeat this until the internal temperature is 125°F for medium-rare and 135°½F for medium. Set aside, tent with foil. Discard the oil/butter and herbs, and repeat this process with the 2nd steak.
In a separate, non-stick skillet, melt 1 tbsp of butter and add the eggs. Cook for 1 to 2 minutes for sunny-side-up. Season with a pinch of salt and pepper.
Place the cooked steaks on plates, and top with roasted tomatoes and eggs. Serve immediately.