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Combine melted butter, soy sauce, and garlic in bowl; set aside. Pat steaks dry with paper towels and sprinkle with 1 teaspoon white pepper and ¾ teaspoon salt.
Heat 12-inch cast-iron skillet over medium-high heat for 5 minutes. Add oil to skillet and swirl to coat. Add steaks and cook, flipping steaks every 2 minutes, until well browned and meat registers 120 to 125 degrees (for medium-rare), 10 to 13 minutes. Transfer steaks to carving board, tent with aluminum foil, and let rest.
While steaks rest, add zucchini, onions, mushrooms, remaining ¼ teaspoon white pepper, and remaining ¼ teaspoon salt to fat left in skillet and stir to combine. Pat vegetables into even layer and cook over medium-high heat, without stirring, until beginning to brown, about 3 minutes. Stir and continue to cook 2 minutes longer. Add mirin and 2 tablespoons soy-garlic butter to skillet and continue to cook until liquid has evaporated and vegetables are well browned, about 2 minutes longer.
Transfer vegetables to serving platter. Slice steaks ¼ inch thick and transfer to platter with vegetables. Drizzle steaks with remaining soy-garlic butter. Serve.