In a small saucepan, stir together all the Teriyaki Sauce ingredients. Bring to a boil over medium-high heat, stirring frequently; continue cooking, stirring occasionally, until syrupy, about 5 minutes. Set aside to cool to room temperature.

In a gallon sealable heavy-duty plastic food storage bag, pour half of the Teriyaki Sauce. Add the 2 steaks to the bag, turning them to coat with the sauce; seal the bag and refrigerate 2 to 4 hours to marinate the steaks.

Heat a gas or charcoal grill to medium-high heat. Meanwhile, in a mixing bowl, toss together the mushrooms, bell pepper, onion, oil, and kosher salt. In a stovetop ridged grill pan over medium-high heat, grill the vegetables until nicely browned, turning occasionally with tongs, 10 to 15 minutes. Retune them to the bowl and set aside.

Grill the steaks to an internal temperature of 130°F to 135°F on an instant-read thermometer for medium-rare, about 4 minutes per side. Transfer to a platter, cover with foil, and leave to rest for 4 to 5 minutes. Then, cut crosswise into slices about ¼ inch thick.

 In a salad bowl, toss the greens with 3 tablespoons of remaining Teriyaki Sauce. Arrange the greens down the middle of individual serving plates; top with tomatoes and blue cheese. Arrange the grilled vegetables on one side of the greens and steak strips on the other. Drizzle the steak with the remaining teriyaki sauce and sprinkle with sesame seeds.