Rub the steaks all over with 1 teaspoon mustard and 1/4 teaspoon each salt and black pepper. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the steaks and cook until browned, 5 to 6 minutes per side for medium rare. Transfer to a cutting board. Wipe out the skillet.
Heat 1 more tablespoon olive oil in the skillet. Add the mushrooms and cook, undisturbed, until golden, about 2 minutes. Season with salt and continue to cook, stirring, until just soft, 3 more minutes.
Whisk the parmesan, sour cream, 3 tablespoons water, the remaining 1 tablespoon olive oil and 3 teaspoons mustard, the green peppercorns and brine, and 1/4 teaspoon each salt and black pepper in a large bowl. Add the mushrooms, lettuce, celery, tomatoes and croutons and toss. Cut the steak into bite-size pieces; add to the bowl and toss. Divide among plates and top with more parmesan.