Toast sesame seeds in a large nonstick frying pan over medium heat, stirring, until golden, 4 to 6 minutes. Pour into a large bowl. Heat peanut oil in same pan, then add a single layer of won ton wrappers at a time and cook, turning once, until golden, 1 1/2 to 4 minutes. Transfer to paper towels to drain.
Combine mizuna, mustard greens, bok choy, cucumber, cilantro, and green onions in a second large bowl.
Add vinegar, soy sauce, sesame oil, sugar, and chili oil to sesame seeds in bowl and whisk to blend. Spoon about one-third of dressing over greens and toss to coat. Add chicken to remaining dressing in bowl and toss to coat. Gently toss ingredients from both bowls together, then toss one more time with won ton triangles.