If it’s 80 degrees or cooler, I take my cast-iron skillet outside and use it on the grill. If it’s hotter than 85 degrees, forget it! You’ll find me in the kitchen with the AC cranking.
Using a meat mallet, pound the steaks to an even thickness of about 1/2 inch. (A few strokes should do the job.)
Let the steaks come to room temperature. Heat a large cast-iron skillet, griddle, or grill over medium-high to high. Season the steaks liberally with kosher salt and pepper from above, like rain. Drizzle the steaks with olive oil. Cook until browned, about 4 minutes. Turn and cook to desired doneness, 3 minutes more for rare, a minute longer for medium, and 2 minutes more for medium-well. Place 1 lemon half, cut-side down, in the skillet. Cook until caramelized, about 2 minutes. Transfer the steaks and lemon half to a cutting board and let rest while you make the salad.
Thinly slice the mushrooms and onion; mix together in a shallow dish. Drizzle with the juice of the remaining 3 lemon halves (about 6 tbsp.). Add the thyme; season liberally. Let stand for 15 minutes. Toss with the celery, fennel, parsley, Parm, and EVOO; season.
Slice the steaks against grain. Douse with the juice of the grilled lemon half. Serve the steaks with lots of the salad.