Heat 1 tablespoon oil in a large nonstick skillet over medium-high. Add broccoli and mushrooms; cook 6 minutes. Remove broccoli mixture from pan (do not wipe out pan). Add remaining 2 tablespoons oil to pan. Add rice and cashews; cook 5 minutes. Stir in eggs; cook 1 minute. Stir in broccoli mixture, 1 tablespoon soy sauce, and pepper. Combine remaining 2 tablespoons soy sauce, peanut butter, vinegar, and 1 tablespoon water in a bowl. Top rice mixture with peanut butter mixture and sesame seeds.