In a small bowl, mix melted butter, soy sauce, and minced garlic; set aside.
Pat steaks dry with paper towels and season with 1 teaspoon white pepper and ¾ teaspoon salt.
Heat a 12-inch skillet over medium-high heat for 5 minutes. Add vegetable oil and swirl to coat. Place steaks in the skillet and cook, flipping every 2 minutes, until well browned and internal temperature reaches 120-125°F for medium-rare (10 to 13 minutes). Remove steaks, tent with aluminum foil, and let rest.
Add zucchini, onions, mushrooms, remaining ¼ teaspoon white pepper, and remaining ¼ teaspoon salt to the skillet. Stir to coat in rendered fat and cook over medium-high heat, without stirring, until browning begins (about 3 minutes). Stir and continue cooking for another 2 minutes.
Add mirin and 2 tablespoons of the prepared soy-garlic butter. Cook for another 2 minutes until the vegetables are well browned and liquid has evaporated.
Transfer vegetables to a serving platter. Slice steaks ¼ inch thick and place on the platter. Drizzle with the remaining soy-garlic butter and serve hot.