Place 2 tablespoons pistachios in food processor container. Cover; process until ground. Coarsely chop remaining pistachios. Set aside.
Combine ground pistachios, 1 tablespoon thyme and steak seasoning; press evenly onto steaks. Brush mushrooms with oil.
Place steaks in center of grid over medium, ash-covered coals; arrange mushrooms around steaks. Grill steaks and mushrooms, covered, 10 to 14 minutes (over medium heat on preheated gas grill, 12 to 16 minutes) for medium rare (145°F) to medium (160°F) doneness and until mushrooms are tender, turning occasionally.
Carve steaks into slices. Season with salt and pepper, as desired; keep warm. Chop mushrooms. Combine mushrooms, rice, chopped pistachios and 1 tablespoon thyme in medium bowl; season with salt and pepper, as desired.
Drizzle beef with some of the balsamic syrup. Serve with rice mixture and remaining balsamic syrup. Garnish with thyme sprigs, if desired.
Nutrition information per serving: 457 Calories; 180 kcal Calories from fat; 20 g Total Fat; 6 g Saturated Fat; 10 g Monounsaturated Fat; 71 mg Cholesterol; 135 mg Sodium; 38 g Total Carbohydrate; 4.3 g Dietary Fiber; 28 g Protein; 4.1 mg Iron; 7.9 mg Niacin; 0.7 mg Vitamin B6; 5.1 mcg Vitamin B12; 9.1 mg Zinc; 51 mcg Selenium;